Calculus 1 Fundamentals: Limits & Derivatives | Tanmay Varshney | Skillshare

Calculus 1 Fundamentals: Limits & Derivatives

Tanmay Varshney, Software Developer, Tech Educator

Calculus 1 Fundamentals: Limits & Derivatives

Tanmay Varshney, Software Developer, Tech Educator

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18 Lessons (2h 51m)
    • 1. Calculus Fundamentals : Overview

      3:22
    • 2. Introduction to Limits

      0:51
    • 3. Concept of Limits

      4:11
    • 4. Existence of Limits

      8:17
    • 5. Examples

      8:53
    • 6. Algebra of Limits

      7:36
    • 7. The L'Hospital Rule

      15:38
    • 8. The Sandwich Theorem

      6:01
    • 9. Limits of trigonometric functions

      8:03
    • 10. Limits at Infinity

      15:36
    • 11. Miscellaneous Examples-Limits

      19:10
    • 12. Introduction to Derivatives

      0:42
    • 13. Concept of Derivatives

      9:07
    • 14. First principle of Derivative

      9:07
    • 15. Algebra of Derivatives

      10:01
    • 16. Derivative of common functions

      11:32
    • 17. The Chain rule

      14:09
    • 18. Miscellaneous examples-Derivatives

      19:09
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About This Class

Master the building blocks of Calculus:  Limits & Derivatives

Section 1 - LIMITS

  • Concept of LimitsFor making you familiar with the notion of the Limit.
  • Existence of Limits: Introduction to the Left-Hand Limit and the Right-Hand Limit.
  • Algebra of Limits: Addition, Subtraction, Multiplication, and Division in Limits.
  • L’Hospital Rule: A handy tool to circumvent the common indeterminate forms when computing limits.
  • The Sandwich Theorem: To evaluate limits of functions that can't be computed at a given point.
  • Limits of Trigonometric Functions: To cover the important limit identities of trigonometric functions.
  • Limits at Infinity: To evaluate limits at points tending to positive and negative infinity.

Section 2 - DERIVATIVES

  • Concept of Derivatives: Meaning of the term "Derivative" in Mathematics.
  • First Principle of Derivatives: The conventional way of computing derivatives.
  • Algebra of Derivatives: Addition, Subtraction, Multiplication, and Division in Derivatives.
  • Derivatives of common functions: To set you free to calculate the derivative of combination of any functions.
  • The Chain Rule: To calculate the derivatives of composite functions no matter how complex they are.

Meet Your Teacher

Teacher Profile Image

Tanmay Varshney

Software Developer, Tech Educator

Teacher

I am a Senior Software Engineer with vast experience of working in top tech giant companies.
I have more than 6 years of industry and teaching experience in domains like:

1. Designing scalable architecture for complex and distributed systems.

2. Developing components in a system across the full stack.

3. Solving complex data structures and algorithms related problems.

These are the major skills needed to be a good software developer who can excel in any tech company easily. I am really passionate about sharing my knowledge and expertise with you.

Thus, I am on board to create awesome technical courses on Skillshare based on my expertise which can be understood in the simplest manner.

Come, join me in this learning adventure!! I will... See full profile

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Transcripts

1. Calculus Fundamentals : Overview: Hi, guys. I am thin Marash. Me and I will be your instructor for this course on calculus one. Fundamentals. So first of all, I will review about myself. I am a software developer and a Mac enthusiast. I have been graduated in computer science along with mathematics. Also, I have eating experience off around three years in mathematics. Now what is calcalist and via I'm interested in teaching Helpless. Helpless is the study off change with its primary focus on two things. Great off change, which comes under differential calculus on Accumulation, which comes under in Eagle. Helpless has helped in reference development in different branches off mathematics, physics, engineering, economics, statistics and many more. So there is endless scope off helpless in our lives. Let's begin to discover unless possibilities. Now let us focus on the prime objectives to be achieved from this course, you will be equipped with the basics off helpless. You will become for table in solving any problem. Let it limits in derivatives by getting enhanced exposure to a wide variety of problems. Also, you guys will be prepared enough advanced calculus forces in future. Now let us focus on how this course is structured for you. In this course, you will be covering the building blocks off calculus limits and the elevators in an interactive manner with help off wide variety. Off examples in grafs Key features. Off the courts are video lectures. Course has over 15 lecture videos, which essentially cover each fundamental topic in limits and derivatives. Next, this quizzes you understanding, developed through the lectures can be tested by taking quizzes introduced in the course at regular intervals. Next is quiz reviews. You can review your answers by taking a look at the detail solutions provided. Even after this. If you need a better level of understanding, feel free to discuss in the Q and A section. Next, we have miscellaneous examples. After every section we cover miscellaneous examples off. The major topics varied across all levels in order to expose you or diversified set of problems. Lastly, we have formally and references which are provided for a quick glance or the formally notations anthems used throughout the course. Now, what are the prerequisites for this course? Basic knowledge of functions, basics off algebra, basics off trigonometry would be a plus to have, however, all of the above prerequisites are optional, and we will be providing a D quit references related to the course for your convenience. So if you want to quickly mastered the fundamentals off helpless, this is the right course for you. Step up with me on this journey full of learning. But this course we are committed to provide you the education you want and the attention your result. So let's enroll, learn and master. 2. Introduction to Limits: in this section, we will be covering the limits firstly with the start, by getting familiar with the concept off limits, then level. Discuss about the existence off limits. Then we will discuss the algebra off limits where we will see how to deal with addition, subtraction, multiplication and division and limits. Next level. Cover the handy a lap it'll room, which helps in calculating the limits quickly for special cases. After that, we will cover the famous sandwich term, which helps in evaluation off limits for complicated functions. Then we will see how we will weight limits off pragmatic functions. And finally, we will see how limits are being evaluated at points ending Go infinity. 3. Concept of Limits: Hi, guys. With this video, we will be starting with limits and derivatives. Let us first have a look, but exactly a limited. So it will be starting with introduction to limits. So I will be explaining you the concept off limits through a basic example. So let's say we have a function F X is equal do for express for over X plus one. So this is nothing but, as you can see, four times x plus one over X plus one. So as you guys must be wondering that you can easily cut off this express one from numerator and denominator. But this is not true. Since at X equal to minus one, this function will be at X equal to minus one. Dysfunction will be four times zero over zero. So this is nothing but zero by zero. And this is in the dominate form. So the function is not define at X equal. Do minus one. Let's try to draw the graph off this function. So this is our Y axis. Um, let me make it straight now. So this is our by access, and this is our XX is this is minus one minus two minus three one do three, one, 23 And for so as you can see that the graph is not defined at X equal minus one and apart from this at every other really off X The graph is nothing but equal before. So the grounds will be something like why I called before Ah, horizontal line and only at X equal to minus one. The graph is not define, so I'm making it open that X equal to minus one. So this is the graph off. Why called affects dysfunction. So now our objective is to find the value off this function at values of X very close to minus one. That is when excess approaching to minus one. So let us now it computes the value of this function then exists approaching to minus one. So let's say the approach X equal do minus one from the left side, that is values off X less than minus one. So we have something like minus 0.1 point 00001 So this is very close to money so naturally , but not exactly minus one. And we're approaching from the left side. So this value will be nothing but for similarly when we approach from the the site that this right side on the value of X again very close to Edison. So let's say we take very close to minus one from the right side and that is minus 10.99999 So this is again equal do for So now you might me having ah picture in your mind as X is approaching towards minus one and dysfunction the I love this function is approaching toe four. So here is the concept off limits. It is coming into the picture that that is, we write it as lim X approaches to minus one F off eggs is so in. This isn't in our cases that collects, exactly could do for since we're approaching from the left side or from the right side, the result is the same. So we can see as X is approaching two minus one. A function is approaching the words for So this is the basic concept off limits that I have explained through this very simple example. So guys do join me in the next video when we covered the topic existence off limits. This is for it this time 4. Existence of Limits: so Hey, guys, welcome to our discussion on limits in this video. We'll be discussing about the existence off limits. So let us see. We have a function for fix is equal to the lesson teacher off X. So let us draw first the graph off this function. So we have this by excess over here, and this is X axis. This is one do three similarly minus one minus two ministry and this is one do and three. So let's rather graph for this function, says Zito. So from values of X, from 0 to 1, the value of greatest indeed, Iraq's will be equal to zero. So we'll be making it open that X equal to one similarly from one to do and similarly from 2 to 3. So this is the graph for greatest in DJ Rex F of X is equal to reduce indigent X right. So now that from our previous video, we have to compute limit at a certain point, So legacy will be computing LTD X equals one. So our objective is limit X approaches to one ffx. This is what we need to l quick for this particular function. So that's no evaluate the value of this function as we're approaching towards one. Let us first approached from the left side, and values of X are very close to one, but less than one. So that will be function. Let's say f off 0.99999 So, computing from this brunch we will see the value will be Tzeitel. So this will be f off 0.9999 will be zero since we are at this brunch. And similarly, when we approach this X equal to one from the right side the legacy we compute the value off this function at 1.1 That is from great safe. So at this value affects the function will be on this branch, so its value will be one. So from known will be taking this value that this ah the value less than that point and very close to that point we will taking will be taking it as one negative and does this one was it? So this is this symbolic meaning off values off X less denman, but very close to one. So this is value affects greater than men, but very close to one. So now it will be riding it as limit X approaches to one my one minus. So as we're approaching from the left side So this is nothing but the left hand limit. So this will be equal. Do zero this weekend. Writers f off one negative. So this will be equal to zero. Similarly, we will be covering. Or we were writing the effort. One positive that is Lim X approaches to one positive ffx. So that will be one. No, we will see that this this will be a little and this is the right tournament. And so this is the left argument. So this eso no, For this limit X approaches to one fo fix. If this exists, then this should be equal to this as well s this so in order for the existence of a limit, it should not be depending that if we approach from the left side or from the right side, the result should be the same. So this is the basic condition which I'm going to right now that the l A chill should be equal to the original. So this is the condition for the existence off limits. This is the position. Okay, So the limit will exist only if these two limits are equal. And both These also should be a finite quantity. And it is irrespective of the fact that the value off that function is existing at that point or not. So this is a basic condition for existence off limits. So now let us how some other notation as well. So this was Ah, this was the a Little little is a Cold War Egil so original is equal to And this was one negative in our case was equal toe f off one positive from a little equal toe a little I and this was except just the one negative was equal to limit expletives to one positive ffx to know I will introduce a different rotation as well for the evolution off limits, let's say for this limit this little as except purchase 21 minus or we can say one negative anarchist he was writing as X equal 2.9999 So let's say we have X equal boots one minus at and this is a very small quantity, but is a positive quantity. So since x approaches toe one negative. Then this edge will actually approach to zero. This actually will be very close to zero. So you will be writing this left 10 limit. Let's introduce. It's over here. In place affects Phil. That will be limit when excess approaching toe o X X is approaching the one negative. This actual approach to zero. So we'll be replacing this. This thing X approaches to one negative with better produce 20 on this FX will be nothing but f off. When? When I said since X equals one minutes, it's you know, this is the left element. So this is another way offer writing the left hand element. Similarly, for the right element, let us see. We have X equals one place. It's and same thing here. This is a very small quantity. So since when X is approaching the one positive this vegetable approach to zero and similarly we can write this right hand limit as limit. When eso we'll replace this ex protester one positive it at approaches to zero and we will write this X s one plus at. So this is another way off writing the like element. And this is another way, operating the great element on the left and right. So what form known it will be considering this as our tradition, this one. So I was just introducing you another way off writing this left tournament as well as the life element. So we'll be stinking toe this notation. This notation will be our consideration. You guys, I hope you might have understood the existence off limit from this video. Do join me for the next set of videos. Thank you, guys. 5. Examples: welcome days in this video. We're discussing some examples on limits, as you guys might be clear off the basic concepts off limits by now. So we will be discussing some examples now. So our first example is we'll be happy. Well, weight limit X approaches to zero mod X over X. Now, this is more next function. This can be. Do you defined US X when X is greater than or equal to zero and minus X when excess less than zero. So this is the basic definition off this morning, so we will open this accordingly. Then we will be evaluating the left tournament and the right element. So by the basic concept off evaluating a limit, we need to check both the left hand limit as well as the right animal, right? So And the thing is, the condition is both the left hand limit and the right hand limit should be equal for the limit who exist at this point. So, first of all, let's evaluate the left pendulum it. So the left hand limit, according to our definition, will be limited. X approaches to zero negative more dicks over x. Okay, so this is like this end up, we can no ah when this Marnix according to the really affects not since excess negative. Here it's approaching 20 but it is the negative. So this more decks will open as minus X, so this will be weaken right easily. Az minus X Rolex. Not since X is not equal to zero, so the systems get cancer and this limit will be called minus one, so the left hand limit will be equal to minus one. Now let's focus on calculating the tenement so the right hand limit will be lim X approaches to zero positive Marnix or X. Now this Marnix will open as positive X since for X values of X greater than or equal to zero, moored X is equal to positive X according to the definition. So we can simply write it as X by X and you can cancel these since X is not equal to zero here, So this limit will come as one. So now we observe that the left parliament is minus one on the right element is spotted Yvonne and since electoral, that is the left hand limit is not equal book on a chilly that is the right hand limit so we can see that the limit does not exist at this x equal to zero. So the answer is limit does not exist that X equal to zero. So this is the answer. So now let's take a look at another example. So our another example will be a little bit complicated? No. So we should start with it. So our example is limit. Let's make it a little trick. Pneumatic the Stein. So this will be limit except just a zero. Our function is X square. Bless Go sign X. So this is our function where we have well over the limit except for just 20 No, we will be going better. Same steps again and we will be developing plus the left tournament and then the light element. Okay, so left element, first of fun. So that would be limited. X approaches to zero negative X square, bless or six. So, in order to make it more here, we will use the another notation this time with the notation. And we were replacing X approaches to zero negative with X approaches to zero. So in that way I will be progressing in this example. So let's say X is equal. Do zero minus etch. Okay, so now for this example since X equal to zero minus set and X is approaching two. Does he don't negative, then this actual approach to zero. Okay, since excess close to zero, but a very small, negative quantity. So this edge will be is very small. Was that the quantity and will definitely approach toe zero. Okay, guys. So now we will replace this X by etch in this left element. Let's do that. So it will be limited at approaches to zero zero. Minus edge. Hold squared less Go Sign off. Zero minus edge. Right. So this will be nothing but limit edge approaches to 00 minus h hold squares at square less Go Sign off my message on this coastline off my message this will be equal to you might be knowing that co sign off minus X is equal to or sine x itself. So we can evaluate this as limit at approaches to zero. It's where Let's go sign. Etch Now, since edges approaching 20 this execrable also approach to zero and this course I niche If it approaches zero co sign Etch will approach to one so this will be zero plus one and that's equal medical store. So the answer is one. So the left hand limit is coming as one. Now it will be calculating the light element. So here it is, like the right hand limit. This will be a witness Symbolically, it will be written as lim X X approaches to zero. Was it do X squared less Go sign off eggs. Similarly, here also replace excellent h So let's say X is equal to zero less etch And since it is a very small quantity or the quantity and that is approaching 20 was it so x Actual approach to zero x is approaching visit about it. Okay, so now we will be replacing X with edge in the limit that will be limited. So now we will be replacing X with etch in the limit that will be limited edge approaches to zero Zito, less edge whole squared the school sign off zero Blissett. So I am using this in addition off it to make this example or against a solution of this example more into do you guys okay, so That's the main thing. So now this will be again Readiness X square. Plus that this will be go sign itch holds word. Ah, sorry. This is not square. This is just I will sign it late. So yeah, this is personage. So this will be equal to this will be equal to zero less co sign zero That will be done so that it again comes as one now left and limit Waas one. And the right element is also coming s one No, since the election is equal to I login so we can see that limit existed X equal to zero An X value is nothing but one. The limit exists this time and it is coming. Get finite quantities and the answer is coming as one. So, guys, I hope you might have understood this lecture. This was it for this video. Do join me next time. Bye bye 6. Algebra of Limits: Hey, days, welcome again to the discussion on limits. This time you're discussing about the topic. Algebra, filaments. Let's say we have functions F X and G affects, and we are well waiting limit at except pledges to it. Let's say this is the point at which we have to well, over the limit on both of these limits. Lim Limit X apologised away FX and limit X approaches to a G affects exists. Okay, so then the limits can be evaluated according to the normal algebra, so we will be focusing on those things. The first is the addition, so I'll be explaining this within example. So let's see we have Ethics is equal to X squared less X and GX is equal to, let's say, Oh, that's FX physical two X Square and G X is equal to X plus one, let's say, and we have three. Well, what limits? X approaches. Let's say one at four. Fix less geo fix and this function is the result off addition off two functions. So how we can even look limit care so we can we assume that this limit exists at X equal to one, and this limit all to exit at X equals one so we can just break up these two limits according to the additionally So this will be Lim X approaches to one FX Let's limit except just one g of X. So this is like we're splitting in the form off edition. So this will be one squared. It's just one less X is approaching one. So explain will also close to one. And here in GX, we will put the limit on the exhale approach to to Sudan's history. So this is the basic example off the addition off limits. Similarly, we can focus on the subtraction. So let's see if X is X quit and g of X is X minus three. So we will have to be velvet limit except watches to one ffx minus geo fix. So this is the bigger function f X minus geo fix. Not since, according to the previous thing, like an addition here. Also, we can split these limits according toa subtraction, since both of the limits our existing at X approaches to one out at affects alternate DX also, so we can split it as limit except pledges to limit except just one ffx minus Lim X approaches to one Joe Fix. So this will be one square that is one minus one minus three that is minus two. So the answer is coming as three. So this is an example off subtraction will were two functions are in subtraction and we are valued in the limit So we can just break them in the form off subtraction and you allude similarly, we can have multiplication. So this is the multiplication and lets it FX is it will do. X Square plus one and G f x is equal to X less 100. It's ah, some kind of monks and I'm digging. So we will have devalued Lim X approaches to do ffx multiplied by G of X. So this is the function on which we have devalued. The limit affects symbol jacks. So now we first checked that this limit expert, just Toto fx is existing and limit X approaches to two gs GX is also existing right? So we can just simply no split it according to the multiplication rule So we can see this thing as Lim X approaches to do FX in tow limit except justo G of X. So this will be you square last one times to bless 100. So that will be five games 100. So this will be 500 so similarly you can have division also. So let's say if X is ah equal to X squared on DA G if X is X minus two So in this thing we have to take into account one more fact that these two limits need to exist. But also like an division. One more thing like the limit is evaluated. Let's say X approaches to to ffx So this affects my GX With this Dfx should also be should not be called a zero at the point because then the limit will not be defined. Okay, so and then FX and its subjects and this as well So the limited is existing. So we can now split it according to the regional here. So this will be a limit except just to tow effects over lemon except just to to do you fix . Now we'll check one more thing like this GX is existing and this limit X approaches to GX should not be equal zero. So this g extra inaudible is 20 at this point. But when we put the limit that this limit except purchased O G affects the answer of this function is coming as zero. That means this limit does not exist. This limit except Justo G of X is not existing. Okay, but this limited limit is not what you think. So we can say this undefined since denominator Kinda zero. So this is the basic, uh, concept of algebra off limits, which I want to lose in this video. So, guys, this was it for this video. Do join me in the next video. We'll recover the hospital room. Thank you, guys. 7. The L'Hospital Rule: Hi guys. We were discussing a new rule for evaluating limits, that is and hope it'll rule. So let's say we have to revaluate limit at some point. A ethics over GXE on this day can be any real number are positive infinity or negative infinity. Okay, so these are the values off a which they can have and this limit X approaches toe a FX exists and limits Lim X approaches to a G off GX also exists. But the result off this overall limit comes as zero by zero Solar Was it a negative infinity over positive or negative infinity. So when we well over the limit and our answer comes like this zero a zero plus minus and radio were plus minus infinity Then in that case, we can solve the limit or evaluated the limit by the hospital rules. Okay, so let's first define what is l capital rule. So this limit effects over GX is our consideration. So using the capital rule weaken right this limit X approaches toe a FX over GX as limit at X approaches toe a if prime eggs over g premix or you can see if dish X over judicial X and what are half dash X over orgy districts. They're nothing but but the derivative off effects and GX, respectively. Now it is Daddy with you. This will see in the next section. But, uh, for no You just remember that f Primex or FDA sex is derivative off FX and similarly g Primex orgy Dash X is derivative off geeks know how we define how well defined. Anyway, deliberative is the rate off change off that function with respect to the input perimeter that this F Primex or F dash X is the rate off change off ffx with respect to X. Just remember this definition for now. Okay to the rulers limit expletives to a affects over jigs. And if it comes to either zero by zero Oh, plus minus infinitely over less minus infinity Then we can't value it. This as limit except justo a If dish eggs. Well, what? Judith? Six. That this We find the dead off the numerator as the less off the denominator function and ever still comes as zero by zero plus minus infinity over plus minus infinity. Then we can again find that anywhere there. Similarly so for the sake of similar simplicity. Let's stick closing a little bit. So this is the L Hospital rule that I have just introduced to you guys. Okay, so let's look at an example to make it a little bit more here. So let's see. This is, uh we have a function. Limmer lim X approaches to the food X cubed minus four X square less for X over there. X squared minus four. So this is our question, and we have devalued limit. Except Justo. So now it's for sport the limits in this numerator function as well as the denominator function because both had the polynomial functions and we can directly what in the limits on we observed that this is it. Minus four times that is 16 4 times toe for that 16 plus eight, that is eight minus 16 plus eight. So this is a zero now in the denominator square with just four minus four. This is the incomes at zero. So we see that this whole is coming at 00 Okay, so let's first approach this. Using the factory ization mattered by making factors on both numerator and denominator. So let's fact praised these boots so we see that if you put X equal to two in the numerator , this party normal becomes zero so that this means that X minus two is a factor of this polynomial. So we can write it as X minus two as X equal to who is the root of this polynomial. So X minus two is a factor of this party normal on If you divide this whole point normal by X minus two, we will be left with X square minus two weeks. So this party normal after textract factories in would be like this on the denominator Could Britain s X minus two into X plus two by the identity x squared minus whites were so this will be limit except protested to also this X squared minus two X can be split into factors off X. So this can be ruinous X and do expense too. And this is X minus two money to playback express to So now if you look at this here excess approaching Toto and exactly not equal to do so right so we can cancel these factors from numerator and denominator. Since X is not equal total, it would if it had been to this would be 00 they So we cannot put X equal to pull here. Since X is approaching Toto, we can easily cancel our distance from the numerator and the denominator. So this whole thing has now become limit except registered to X multiplied where X minus two over X plus two. So when we put the limits, this will become as two multiplied by minus two, which is zero well water for So the answer is zero. So but we have done this with the help off victimization. Okay, Now let's try to use the l hope it'll rule here, So let's actually help the loot. So if you put X equal to do Oh, in this we get zero by zero from here. So which is an indeterminate form on? This is our this condition for the l Hospital rules so and we can apply no and hope it'll julier right. So let's try to implement the trouble in this. Let's ride the hospital rule a rule again if Lim X approaches to zero. So let's a FX cooler. GXE is ah, formal zero by zero. This is the case in our example so we can simply write it as Lim X approaches to a F Primex over G. Primex. So let's write our example again. They may accept protested to X Q minus four X squared, less forex over X squared minus four. So now let's differentiate what the numerator as well as the denominator So we can write so different station off access to the power and is en times X rays to the power and minus one . But this is a rule of differentiation between. We'll see later in that elevated sections of the scores, you will see this so we will be discussing their and mortgages. So as you know, you just remember the simple rule that the word of off excess stood apart and is nothing but an were deployed by extra strength power and minus one. So what this rule will be differentiating the numerator as well as the denominator, so x cubed, different station will be three times X square minus four. Uh, you can take it out of this four as a constant and derivative. According to the rule for X squared, this will be two times multiplied by X rays to the power to minus one. So that is one less for my supplied by the weight of X will be one. Okay. Similarly, in the denominator, it will be two x on different station off A. A constant with respect to X or any other variable is zero. You just need to remember this for no. So the different station off this constant, that is four will be zero. So this is the differentiation off the numerator and the denominator. So I'll write it more clearly. No, this is three X squared minus eight x less four over Blix. So now let's put our limits again. So now our numerator will be three times for minus eight times. Do plus four over for So this is one in minus 16. Less food over four on this is zero and also it's not an indeterminate form and this answer is acceptable. So the evaluation off this limit is coming as zero. So if you observe the difference between this approach on the previous approach where we use the factory ization matter, they're So we're breaking the people normal into factors was time consuming. That's Andi that required some hard work, took accurate the limit using this. But if you use the capital rule toe, evaluate the limit, then it can be really fast. You just need to know the basic rules off definition over your different station. So this, uh and you can easily devalued elements. So this is a little short hand technique who were over the limit. So let's now take a look at another example on this. So our question is, then my ex approaches to a extra S to the power and minus eight is to the power and older X minus eight. Let's put the element X approaches to weigh in this function, it will give us a to the power and minus arrested Barton it zero minus eight to the party in the zeros of this and you know my name Denominator. So just coming at zero. So this is for more zero basically. So hence we can apply the capital here, right? So let's apply that So excited the rule again, it's ah Lim X approaches to a FX over GX. If both of these limits our existing but the numerator and the denominator and our evaluation is off the form this its effects of objects and the evolution is resulting in in the dominate form many poor delivers. Then we can apply capital here on we can say limit X approaches to a F Bram X over G breath mix or have defects over JI districts where the FBI mixes derivative off effects and G Primex is derivative off GXE, right, So Okay, no, let's find declarative off both the numerator and the denominator. So f primex will be as discussed in previous example delivered off access to the power and is end times X rays to the power and minus one. And, uh is a constant. And hence it will result in zero as a derivative offer constant this feel or what they would do of excess one. One times you can say excessive power one minus 10 So it will be one in clicks to the power It's confessed one minus zero as a zarina constant here. So this comes as limit except bridges toe a end times extras to the power and minus one and now we can easily substitute X equal to a. In this expression on, this becomes end times it is to the power and minus one substance on baby since that we can substitute X equal to here. So this is very easy. Way off evaluation off this limit and contracted this if you would have applied defect transition method well over the limit. This limit then on the numerator. Ah, as a factor X equal to a and you could have also cancelled the X minus eight. I'm from the numerator and the denominator, but it would have been very tedious to calculate the demanding factors off the small Normal . So it will take a lot off. Bang. Actually, guy cleared the end reserved for the summer. Right? But this hospital rule this rule enables you to get the dessert really fast. So I guess it took us let less than a minute to get the answer using this rule. So if you know just the same basic rule some basic rule of derivatives. But you can either. Right? Usedto you haven't read your result or use it. Very fiery. Verify Reserved Break. So, guys, this was it for the hospital room. Do join me in the next video where we discuss about the sandwich term. Thank you 8. The Sandwich Theorem: Hello, guys. Welcome back In this video we will be covering sandwich to him. This is also known as the squeeze term. So there to start on this. Let's say there are three functions ffx, geophysics and traffics. These three are the main functions so that ffx is always greater than equal to G affects which is always greater than equal toe edge affects on this is true for all X and neighborhood off Appoint a where is any real number? So if this is the case, then along with this if limit except riches toe a better fix is finite and it is equal to a value l let's say on along with this limit X approaches toe a ffx is also equal to same value. And in that case, Lim X approaches toe a G affects will also be equal to the same limit that this and okay, so let's make it a little more year with the help over diagram. Let's see, This is our function. Why physical toe fx FX on. This is why is it called toe edge affects function and this is why is it going to be affects in between off this function in the neighborhood of this point. So this is the neighborhood off A and we're just worried about the neighborhood off A. This eight can be anybody. So this dfx is sandwiched between ffx n g effects in the neighborhood of this 80.8. Okay, so this is not a big self intuitive that if limit off dysfunction at some point A is a finite value, I don't know. Similarly, this is also a limit of this function at To fix is also existing when this function is approaching. In the neighborhood of this point, a values also l and limit ex operatives toe a GX will also be equally toe. And so it is more appear from this graph. Let's Okay, so this is the term. So let's take an example to make it more here. So no, it's take an example off sandwich, too. Let's say there is a function Lim X approaches to zero ex course one by X. So the function is X course one by back when the X and we have devalued the limited X equal to zero. Okay, so now this function goes off one bags is a little vacuum function, and it's not in duty in orderto well over the limit off this function. So even make use off the just now introduce sandwich to him, that is, we will find all functions in which this function is sandwiched in the neighborhood of this point X equal to zero. So let's try to do that. So, as we know course, sex is always lying between minus one to or sort of okay, since its ranges from minus one to was a big one. Similarly, this cost one by eggs will also lying between minus one and was a different since dysfunctional in just from minus one. Was it one? So they put Miami Desert and make any difference. But breuder, this one by X should always be defined. Since that X is approaching 20 here, it is not equal to zero. So that's why this one by X will always be defined. And this course one backs will always range from minus one to positive. Okay, so now let's multiply all the three sides with my with positive X in order to make the function similar to the question so that would be minus X is less than or equal to X course one bags, which is less than it will do. X. You know, let's say this function is ffx. This function is G of X, and this function is edge affects, as you can see in the neighborhood off X equal to zero dysfunction. Do you fix a sandwich between FX and it affects? So by the sandwich storm, you can see that the limit except purchased a zero F effects will be equal to limit except lo Justo a zero and to fix. And that will be equal to limit X approaches to zero g of x. Okay, so let's first evaluate this limit to limit X approaches to zero f of X is minus X. So if you black in Lim X equals X equal to zero. Nothing's here except approaching 20 so we can easily like an X equal to zero kind minus X will also a pro Susie Rosen's excess approaching 20 so this will be zero. Similarly, we can compute this thing that the limit, except pledges to zero at effects, will be limited expletives to zero was it detects. So this will also approach to zero now, since limit except just a zero. Do you affect the same as limit expletives to zero. It affects. This implies that limit expletives. Ozio Jfx. That is a function given in this question. That is ex course one backs. This will also be equal to zero. So this is our result. This is the answer to this example. Okay, so we now medios off the sandwich term in order to compute this result. So this was it for this video? Do join me next time. Thank you. 9. Limits of trigonometric functions: Hey guys, they come back in this video, we'll be discussing a new topic that is limits off pragmatic function. So let's start with our first identity. So our first identity is lim X approaches to zero sign X by X. This is equal to one. So this is our first identity on we will prove this identity No. So by the expansion off sign ICS, we know that sine X is equal to X minus X cube by three factorial less extra depart five by five factorial and soon. Okay, so are like, let's straight one mortem that would be acceptable are 7/7 factory The soon so now we will divide the sign ex my ex So that will be Sinus signings by X is equal to one minus x squared by three factorial less extra the par four by five factorial minus X to the bar six by seven factory and soon. So now we can easily like in X equal to zero so that we can evaluate Lim X approaches to zero. So that will be lemon X approaches to zero sine X by X. Let us write it more clearly. That would be sine X by x it's And now you can plug in here to the value of the limits so X equal to zero we can plug in in this expansion. So that will be one minus X squared by three factorial sex for by five factorial minus extent of our six by Sanford grandson. So now you can easily broken X equal to zero. So this, uh, this will be zero. This will be zero and all other times will also be zero since X is approaching 20 x quibble approach to do you know and all all the higher times Well, obviously opposed to zero. So the answer will be but two hands. We have proved that limit expletives to zero sign X by X is equal to one. So this is our first identity. Okay, guys. So there are some more important results as far as pragmatic functions are concerned for limits, So the other one is limited. X approaches to zero co sign X is equal to one. So this thing might be here to you as it is very into do that when X approaches zero go sex approaches to one. So this is the second identity for trig pneumatic functions and limits. Now the third identity identity is limit expletives to zero annex. So this is also equal to this is equal to zero on. We will be doing this now So as you know, that panics is nothing but sine X by Koszics so we can write it as unacceptable to sign Expect coastline X So we will a multiply both the numerator and denominator by X So this will be eccentric or six and this will be extend to signings. And this thing in there is limit X approaches to zero Sign expects is nothing but fun one right according to the first identity. So this weekend, right rest one into lim X approaches to zero. So by the algebra off limits, we can write the last one multiplied by Lim X approaches to zero X by 46 Right, So when Exposition zero Uh, this course I next we'll approach to one And the sex well, obviously approach to zero. So this will be one multiplied by zero, divided by a finance number which is which is approaching to one since cortex will approach tour and okay, so this will be nothing but zero. So the reserve is coming s zero So the limit expletives to zero annexes Nothing but zero. So now we shall use these identities in some examples, Toby more family with how to apply these identities. So let's say we have in our first example said, Let's say we have to value it. Um limit ex separatist 20 one minus course X over X. So this is our example, right? So we're evaluating this West, Uh, we need to be. I fear that there is an identity one minus cost. Two x is equal to do sine squared x. Okay, So if one minus cost two x is equal Toto Science Kerekes, then one minus Koszics will be equal to who signed square X by who as well just have the angles here. So let's put this thing in the question so we will rewrite it as limit expletives to zero to sign square expert do over X. So now where? What are we going to do next? So this thing is Science square expert expert, right, So it's identity can be used from the previous news agencies. Just try to recall. So I think you might be able to recall that we can meet make use off the identity Lim X approaches to zero sign Xbox is one right, So we will make use off the identity. So let's take this thing square thing into consideration. So let us try to separate it in the form of sine X by X. This is sine X by two over expecto and this thing's whole square So expect will hear if we multiply in the numerator and the denominator as well. Right? And now look at this thing Lim X approaches to zero sign Expecto over. Expert do well opposed to one as you can relate from the first identity So we can Then we can make texting as well on this whole thing will become limit Exports is 20 to buy X multiplied by one multiplied by X square by four since this was expect to hold square. But this is coming as limit except sister Zero expert too. And now we can plug in X equal to zero in this. So this will come as zero. So the answer is zero right? So I hope you guys might have understood the basics off how to deal with limits and pragmatic functions, and we will be covering the remaining concepts in upcoming videos. This is it for this video. Bye bye. 10. Limits at Infinity: Hey, guys, welcome back. In this video we will be covering the last important topic off limits limits on X approaches to also toe and negative infinity. So we have reevaluated Lim X approaches to infinitely ffx on Lim X approaches toe minus infinity ffx So we'll become covering examples like this in this video. So let's just try to start with an example here so we will take our first example. So we have limit at X approaches to infinitely extra The bar four minus X cubed less six X square less eight x over five x to the bar four plus two x to the power three minus four. So now let's try to Blufgan X equal to infinity in the numerator and the denominator as well. So this will be infinitely minus infinitely last infinitely plus infinitely similarly in the denominator that will be infinitely minus infinity plus four or minus work. All right, so this thing minus infinity dissing infinity minus infinity. Just want this thing. This is indeterminate. We can't compute infinity minus infinity. Had it been come beautiful, then also, this numerator would have been infinitely just assuming Espen office had been computed then also, if it will come as infinity in the numerator as well as in the denominator. So this thing is coming as indeterminate from so anyhow, if you try to plug in X equal to infinitely here it is coming as an indeterminant form. All right, so now what did you So whenever we have to evaluate a limit at X approaches to infinitely, whether positive, infinity or negative infinity a busy thing, we have to keep in mind that there we have to take out the largest power off X from the denominator. As a factor from put the numerator and the denominator right to the largest part of X from denominated as a factor, you let spurs take a look. In this example, the largest power off X in the denominator is extra the power for So we will be taking out this as a factor from the numerator as well as the denominator. So let's do that. So this will be limited X approaches to infinity. Let's take extra power for common. So the new monitor will be to minus one by eggs less six by X square, plus eight by X cubed right. Similarly, in the denominator not repeat it. And as extra, the par four on inside it is five. Let's do buy eggs minus for Oh, sorry. Uh, because it is Dubai X only, right? And this is minus for by extra the bar for right, so And this export can be canceled. Sin X is approaching to infinity. It is a last number, but not equal to infinity. We can cancel out extra depart for from the boats, the numerator and the denominator. So now you might believe you fear by where did we did this? We have done this because in the remaining part of the numerator and denominator, now it's quite easy to plug in X equal to infinity. Since when? It's except not just to infinity. This one by ex This thing one bikes will approach to zero on this thing six by X square. You also again approach to zero. Similarly, these to these things all these parts So these three parts and these two parts here will be approaching 20 Right? So this thing we can relight as to minus 00 Placido, that will be 0/5 plus zero minus zero. That would be again asked. The answer comes us totally fine. So this is the way to compute limit on X approaches to infinitely in any function. So this is the way what we did. This is the key point to remember that we have to take out the largest flour off X as a factor from off from the numerator ended a normal off the new tenement, right? This was the case when we took into consideration except register infinity. Let's take an example the next separate Just stoop minus infinity, right? So the question is lim x separatist to minus infinity. Let's say things time. Take another variable. Let's say, is that a purchase toe minus infinity four ZEC square plus to the power six over minus two Zed to the power six plus three There. Right, So now here also what we will do We need to take out the largest power off denominator as a factor from both numerator and denominator. Okay, so let's do that. So we will rewrite this as limited set approaches to minus infinity. So what is the largest party in their denominated will be taken Now it's effective. So this is the good of our six. So this will be a factor from both numerator and denominator. Okay, so let's trick. Try to take out the truth of our six as a factor that will be sent to the bar six on four by said the bar four left one over there to the power six into minus to plus three buys their to the power fight on this thing that we are six can be cancel out since exit is not actually not equal to infinity. So this thing will be no limit said approaches to minus infinity four bisected the bar four last one divided by minus two. Last three by said to the bar five So now let's look inside equal to minus infinity in this think so now this since that approaches to minus infinity, this set to the par four will approach to infinity. And so this little powerful with a postman, Freddie. Right, So one buys the toolbar for well opposed to see you. So now this thing will come as for into zero plus one over minus two less three. And do not since their produce to minus and pretty Here is it to the par five appeal of also plus two minus inability. Right. Since it is our power, it was even power. So the sign what he was. But here it is. Our power is that to the power of five will also be approaching two minus infinity. No one buys that to the par five will air approach to when by minus infinity 1,000,000,000 approaching toe zero right one by minus introduced legal again The answer will be coming as this will be one by minus to So that would be my son by two. So the answer is equal to minus one by So these two. But the examples when we were evaluating the limits at infinity or we can say that we were evaluating limits that points approaching toe What's it to infinity and negative infinity? So now we will take a look at another example where we will be evaluating limits at excellent just to positive and negative infinitely in the same example. So let's take such example. For example, is the first part office It is limited, except just to infinity. You do it. Ah, Ruler three X squared Last 6/5, minus Luigs. Okay. On the second Vitus limit the same. The same thing is here, but the limit X approaches to negative infinity. So we are taking the same example for both the positive on the negative. Infinity? Yes, right. So we will see how the answer really differ For what businesses? So long. Let's first evaluate for the first part, as we have discussed in the previous example that we have to take out the largest power off X in denominator as a factor from both the numerator as the less the denominator. But you see, in the numerator everything is under squared and in the denominator, the largest power of X is one this one, right. So we have to not take out excess a common factor from board the numerator as well as the denominator. But in the numerator it is under square. So under a square root, taking out a factor off X squared will be equal in taking order factor off X. So we will do the rearrangement now. So long. Let's take out a factor off X square from the numerator in order to take out a factor off ext effectively from the numerous rate. So we will write it as root over X square and go three plus six by X squared on in the Duma . Treat is very easy. It is like X multiplied by five X minus two. So now this x square we will take out from the square root park. So now this thing should be positive was possible So we will take it out as more decks. Since this factor is coming out from the square root, it will be a positive quantity. Please, not this thing. That's and the remaining part will be three plus six by X square under a square root, right. And then the denominator is five by X minus. To make a black backs right are the main thing that pretty for our answer with X approaches to positive and negative in pretty respectively, is distinct marks on since it will open differently for Bothwell. So this thing and this thing will evaluate to zero since X is approaching to infinity or minus infinity then also won by X And when my ex square What this thing will approach to zero. But these two parts will be the same for both the cases on Lee this part more The X by X This part will differ, right? So now let's take a look first for the positive infinity guests. Since positive infinity is a very large positive contradict. So this more dicks how we will open it will open as was the defects. Let's open it. But the guests X approaches to infinity. So this will be x my under route three plus six by X square by, uh five x men Clever five batsmen stood on this X can begin super the sex So the finally we can plug an ex approaches to infinity Weaken Let's play how now Except just to infinite into this spire the final remaining artist This so this thing will go to zero. This thing will also press zero So it will be on the road three by minus two that the Ministry of Tibet So this is the answer when we when the X is approaching to positive infinity Okay, so now let's take a look The next approaches to negative infinity but will be around So the second part is lim X approaches to minus infinity route over three X square Last 6/5, minus twigs. So let us now take help of the previous part. L a starting point toe little this point. It will be same, right? So we can take help of the previous part and we can write it as Lim X approaches to minus infinity. This will be Mondex in new three. Bless six by X squared Cooler X multiplied by five by X minus. Do right, but no. Here this morning's will be negative. X since X is a very large negative quantity Great toe. This morning, X here will open as minus X so this thing will make that difference here. But this thing will be minus X over X and this will be ruled over three plus six by X square over five by X minus two So this X will begin to buy this X So and finally, we can now plug an X equal to infinitely here. So this part will be zero on this part, Will also busy wrote. This will be minus root three over minus two. So this minus and this minus gets cancer. And the answer is rude. Three by two. So we can see that when X is approaching was a different 20 answered game as minus road trip. I do. And when X approaches negative infinity for the same function. Only the answer came as a rude three back to SoHo. Guys, you might have understood how to compute limits when X approaches. Positive or negative? Infinity. So this is it for all the limit concerts. Now, we will cover the miscellaneous examples in the next video. Thank you for watching. 11. Miscellaneous Examples-Limits: Hi guys. In this video we'll be covering some miscellaneous examples on limits. So let's start with our first example. Not suppose ffx is defined by this definition. That is a plus B X. When X is less than one, I guess four. When X is equal to one and it is B minus X and X is greater than one. Along with this, if X approaches to one limit, ffx is given us f off one, then find the value off A and B. So this is our question. Now let's start with the solution. I just think how we will approach in this problem since this limit except just the one FX is given us f off one. Then what is that for one. So you can see from here it is equal to four. Okay, so now this thing limit, except we just tow one ffx. Since it is a little of finite number, that means it exists for sure it is equal to four. So in our case, that means the left hand limit or ex approaches to one as well as the right hand limit for X approaches to one will exist, right? So now Let's first compute the left. 10 women when excess approaching the one so that will be lemon acceptances to one negative and full fix. Right? So no, Here one negative means the value of X is less than one, but very close to one. So which definition will satisfy from these definitions? That will be this one since for X less than one we egg this definition off ffx. So this definition we will take four values affects less than when all we can say one negative so we can write it as ffx is equal to a plus B X here, right since X is approaching the one negative. So the approximate value we can plug in, then this is one. So we will plug in one care, so that will be a plus B. So the left hand limit is coming as it crispy lettuce market. That's one. Now the beauty will compute the attention so the right hand limit will be limit except adjust to one. What's it doing? FF IX Now, in this case for X approaches to one positive we have X is greater than one, since the value of X is approaching one. But it is created than one, so their definition this definition will be our consideration. So that will be, ah, this really B B minus X. So we'll write it as the medics approaches to one positive B minus X right. So this will be now we can plan and execute one, since it's approaching toe one so approximately we can plug in one. So now that will be B minus. A. Don't let this market has to. But now, for the limit exists, the natural should be equal toe theology. And since it is given in the question that also that the limit is existing and it is equal to finite quantity on, that is for so the both the left hand limit on the right hand limit will be equal. So from these directions we can write a plus. B is equal to B minus a, and it is equal to fo phone number. This F one. This is nothing, but it will do four, and so we have two equations with us. A plus B is equal to four. Let's market as three and the B minus A is also equal to four. Let's market us four so Now we have to solve these two equations long lettuce soul in order to find the values off A and B. So from a question number three we have a plus B is equal to four and a plus B minus e is equal. Do for two legs righted as minus a plus B. And this is also for right from a question number four. So I'm just rearranging for making pair off and be right. Okay, so these are our two equations. So knowledge sold this. So we have to evaluate one variable from here in order to find the value off the other people. So let us first add these two equations So when we add this minus, it gets cancelled. With this A and we gets added with this be so this becomes to be equals it. So if Toby is equal to wear, this implies that b is equal. Do for so we have found the value of B is equal before No. So how we will now find out the value of a well, like in the value of being any of these equations. So now let's plug in vehicle before in this equation. So that is a plus B is equal to four. That would be a plus for equal do for, so this means is equal to zero. So hence we are able to find out devalues off a and B and our answer is physical before and is equal to zero like so this is our answer. Now let's take another example. In this example, we have to evaluate Lim X approaches to zero route over one plus X minus one and this one is not in the root on this whole thing is divided by X. So we have developed limited except righteous to zero so long like a straight up like an X , equal to zero in the numerator as well as the nominee. So if you blacken so our answer will come as root overburdened Placido minus 1/0. So this is Rule one is one and mind minus 1 0/0 So this is zero by zero. It is an indeterminate form, as you might be knowing now. So this is indeterminate. So how we really value it. So here one thing we can do that we can multiply, we can multiply in the numerator and the denominator with the conjugal it off this numerator. Okay, so that's tried to do this once. So then it will be more obvious to you that why I have been this right, So the conjugal will be ruled over one plus X less one. So the sign between the problems will you get reversed. So in order to find the congregate So now let's might bet this whole with route over one plus eggs minus one. So we went for sex plus one in the numerator and the denominator. So this will come as Lim X approaches to zero ruled over one plus X minus one multiplied by its consecrate. Right. So that will be on the route one plus X plus one and the denominator. It will be X multiplied by under route one plus X plus one. Okay, are you can see an identity here. Let's say this is a and this is B. So this is a minus B. And this is a plus B. So by the identity, a Myers be multiplied by a plus. B is equal to a square minus B squared. So this is the identity. So we will use this identity in order to make this simpler. So let's do that. So this will be limit. Except it's 20 A square that is the square square will be worked off one plus X holds court . That will be one X minus B squared. That is one. So, uh, over eggs. When you replied back, ruled over one plus X plus one. So this one gets canceled with this one. And this remaining part, which is X gets canceled with this X since X is approaching 20 is not equal to zero. So we can cancel this X to The remaining part is Lim X apologised to 01 over under Rule one plus X last one. So now let's try to plug in X equal to zero. Here. If you blacken X equal to zero, this will be ruled over one plus zero that is one on. So it will be one over one plus one. So that could be 11 by two. Now, here able to get the answer, that is one. By do so, just by multiplying the numerator and the denominator with the conjugated off this thing, this new motor were ableto sold for the limit. So had this thing being in the denominator. Also, we still we still would have done the same thing, right? So this was the example when we had to multiply with the consecrate in order to make our example. Computer bill for limits, right? Let's start with our third example now. So let's say we have a function. F X is equal to reticent teacher off car sex, and we have to find Lim X approaches to I do ffx. So first of all, I will make your understanding more clear about this greatest and teacher expert. So they say this is not cost six. Let's assume. Plus, we have another function. Let's say G X is equal. Do greatest. Let's say G X is equal to greatest in Egypt X. I'll make her in this train here about this greatest into your expert first. Then we will go on a little computer. The limit right to the graph of Greatest. Indeed, it X is something like this. So this is like a Steph Manship. Great as you can see. No, let's say we have to compute creators and teacher off one point to the so let us take this value, and this really will be some somewhere playing here. So the vie corner will be one to the greatest and teacher off. 1.23 will be one. Similarly, let's take a greatest indigent off minus morning, three d So for this, the X value will be staying somewhere here, and the Y coordinate will be lying on this branch, so that will be minus one. So the greatest indigent off minus 0.3 does will be minus one. Let's take another example, like minus 1.5. So minus 1.52 Ah, nice. And we're here, and the bike ordinate will be on this project and that will be equal to minus two. So this one's for giving you the basic understanding off how the greatest religion of X behaves. So now let's try to draw, or you can say, let us try to compute this limit X approaches to I do. Now this pie isn't radiance. So bye bye to will be quill and do 90 degrees. So let us try to first photograph off. F X is equal to classics. Okay, so yeah, First, I will draw the graph off course six So course Texas plotted something like this. Oh, something like this. Okay. And Ah, this is Tzeitel. This is spy battle. This is by this says three by, uh, this is street by about two. And so and this is minus vibrate right on this part. This vie coordinators one and this bike ordinaries minus one. Since core sex will always ranged between minus X minus one was it? So now we will proceed toe compute the limit expletives to bye bye to ffx. And FX is greatest in region or sex, right? So now, in order to find the limits in order for the limit exists, we have to make sure the left hand limit at this point is equal to the right element at this point. By the concept of existence off limits we have to make this point here. So let's say for us we compute the left tournament so the left and limit will be limited X approaches to bye bye to negative ffx. And so it will be limited support just too. Bye bye to negative greatest in digital Koszics. Now let's see from the graph expert just toe. Bye bye. Don't negative means value of X, closed toe by toe but less than pirate. So we will approach from this site. Okay, so in this thing, the value is if we see from the graph the vie coordinate. The value is very close to zero, but it is a very it is a positive one. Dirty right. So let's say this is a very small quantity. Let's Zetsche and we can represent this weekend represent from the graph. But it is a very small country. Let's say that this, uh, equals edge, right, So this we can replace this as well, since we're using a chair. So you can say that is approaching 20 and it is a very small, positive quantity. So the greatest in teacher off this this thing we have to compute. So he s a limit. X approaches to see what a great, distant digit off that we have to compute. Let's say we take a very small well, you let 6.1 or 0.1 Any small value So the greatest in teacher we can compute from this branch the value of wretch like somewhere here So the greatest teacher will come as zero. That doesn't limit X approaches to zero Greatest in digit off edges coming. US. Tzeitel to the left and limit is coming as zero. Now we will try to compute the right and lemon. Okay. Ah, so the let's compute the right element. The right element that is the original will be equal. Do limit X approaches to buy by two. Does it do great distant digit off Koszics Not from the graph. As you can see that when expletives took bye bye to positive that means we're approaching from somewhere here. So we had approaching from this site. This site right on the value of X is very close, but greater than vibrate. So the value will somewhere like here somewhere. Okay, so the Y coordinate will be less than zero, but greater than minus one. And the bike ordinate will lie somewhere here somewhere in this region. So it will be less than zero, but greater than minus one. So this thing we can rely does, since it is a very small negative pointedly, let's say it can be presented as minus edge. It is. It is a very small part of the country. So minor sets will be is very small. Negative quantity close to zero. So are y coordinate. We can represent by this eso we can replace it. Place this whole limit as limit at a protester zero Gaeta Simply Jenoff minus such So this thing Well, we deduce from the graph. OK, so now let's take yours off These examples to compute Ah greatest indicate off my sex These examples here. So when it is very close to zero right, let's say we have minus 00.1 or minus 0.3 Do like hair or minus 0.0 to somebody like this. So something like that Okay, so are extraordinary. Will lie somewhere here, right? Some bit here are by coordinate Corresponding to this X value will be minus one since the Y coordinate comes as minus one. So the limit the distant region off minus edge An edge is approaching 20 will be It will do minus one. So this is coming s minus one. Okay, so now the left hand limit was equal to zero on the right hand. Labor is coming as minus on our since the left hand limit. Onda the left hand limit on the right Handelman. The boat are not equal to each other. And we can say surely that the limit does not exist. So thank you guys, but by 12. Introduction to Derivatives: in this section we will be covering the river does we will start with explanation off the notion off derivatives. Then we've built over the first principle off derivatives. Next we will see algebra off derivatives which lets you deal with addition, subtraction, multiplication and division in derivatives. After that, we will see what are the derivatives of the common functions used and guiltless. Then in the last, we will discuss the in famous chain rule. This rule actually lets you calculate the derivative of composite functions, no matter how complex there. 13. Concept of Derivatives: Hello, guys. Welcome back in this video we will start with the new section that is dedicated. So let us just have a basic introduction. What, exactly? A derivative it's are. Derivative is instantaneous rate off change off a quantity. So in mathematical terms, we can say it is the amount by which of function changes at a particular point. So let's first make it a little more clear with a basic example. So we're taking example off. Let's say there is a function dependent on time on dysfunction is giving us the this from the function is giving us the distance traveled, which is dependent on right. So this is giving at this distance traveled with respect to or we can see um, it is distance traveled as a function off time, right? So let us bigger, basic gruff. Let's see, this is the function. Let's say this is a 50 right, and this is the distance travelled that it's s and this is the input perimeter that dusty. So now we have to compute instant in this rate off change off distance with respect to pain at a particular point. So let's say the point is this one Okay, right. So we have to compute the derivatives off this s dysfunction, etc. I just since traveled with respect with pain. So how we will compute with the rate of change off this function will be given as let's see , this is the first point. So this is the point at which we are calculating that rate off change. And let's say there is a very small change in financial change and considering the next point, let's see, this is the next point. And this is this. Coordinators X coordinate is expert a passage. And if this is a, this will be FAA on. If this is a play set, this will be f off a plus. Etch right. And since it is instantaneous, instantaneous rate often means the tinge is in the input perimeter is very, very small. So this edge is actually very close to zero approaches to zero. So now lives. Let's try to write it mathematically, how we will write that the revenue off this function with respect mine mathematically. So that's straight that so we can see anywhere off the function F 50 a two point e A. So that will be that will be equal to the rate of change that is nothing but a change in the distance traveled. That is F off shapeless edge on this effort. Fee over changing. Then prepare Limited that this time. So that will be a Pless EJ minus. Okay, so this a gets canceled with this A and this remains us f off a place etch minus f a over h And since we were considering very close to zero So now the concept off limit also will come into the picture. So we will say just tending bar zero. So we will write this as the limit at a protester. Zero f off a plus H minus f off a over that. So this is the derivative off softly at easy will do a Also there is one more way of denoting the Delaware mathematically, that is f prime Be so this is steep point. This is at the point equal to a So this is a mathematical be off writing network. So this is the derivative of this function fft that easy will do it right? So also this limit hair should always exist. It always exists in order for this day with you to exist, right? So this is the main consideration. Or we can say this is the basic formula off for computing the derivatives off F off D. So this thing, this thing is what we introduced as a basic definition off computing the anyone. So let us take a simple example. So are example it that we have to find did very well off a function. Let us see a full fix. We have to find a way to go function ffx Zeke will do X squared plus one at a point x equal to three. So remember here, and we calculate a little bit. We can will it at a point. So this point is X equal three here. Right? So now, as we discussed fuming moments ago the basic definition off half to compute the derivatives . Let's abide by definition only and let us compute the river off this function at this particular point. So let's go ahead with this. Do all right? The decorative off f x at X equals three. This notation was F prime X at X equal to three. Well, we can write it as f prime three also. So this is a more short words, you know, fighting this so weaken, right. I've print three. It will be equal limit Etch approaches to zero. So we are taking an interval which is very close to zero. So that really element X approaches to zero f wolf. He passage for a presage. It is three. So it is that that will be minus f off three. Right? Or what? A place that's minus A. That will be great. So how we will compute off three passage for three playsets We can compute directly looking in three plus etc. Into this function. So that will be clement. It approaches to 03 plus edge whole squared plus one minus f of three that this feet square plus one Onda such will remain the same. So now let's open this three plus edge, whole square. So that will be it. Square plus nine. That's two way We that the success plus one minus nine minus one and this one gets canceled . This one on it is so this nine also gets canceled with this name. Okay, so I'm here at this coming up Common. Okay. So people take out that is a factor here and Since it does not be called zero, it is approaching 20 so we can cancel out one edge from what numerator and the denominator . And the remaining is limit edge approaches to zero edge plus six. Okay, so and since it is approaching 20 we can plug in at just zero here, and the answer will be zero plus six. So this will be six. So what is this six Now this quantity six is the very video off this function at X equal three. Or we can say the rate off. Change off this function F X with respect to X at this particular point acceptable toe three that it's six. So this is our answer. So this was it or this video Do join me in the remaining negative videos? Nobody. 14. First principle of Derivative: hello ways. Welcome back in this video we were discussing the first principle off the rebellion. In the previous video, we talked about the basic definition off derivatives. So going by that discussion, only we will be describing the first principle off. So in the previous video were discussed the basic formula for calculating in the river. So for starting with the first principle of derivative will be considering a function. First, let's see ffx, which is a real valued function. So this is the real value function and the derivative of dysfunction By the first principle of derivative will be f premix. It is equal. Do limit it approaches zero f x plus edge minus F of X over at now. If you remember, we were calculating for a specific point like f prime A So where were we were calculating? Like F prime A is equal to f off a press. It's minus four feet over rich limited to protesters zero Of course, this hit is like our general formula. So for it is for any General point X So this is F prime X, which is the radio off ffx. Right. So this is the formula for the first principle of Delaware for calculating the derivative of a function. So this is the formula which will we will be using in our future examples. So if you're differentiating to basically first principle, right? No, some notations are there. This f Primex is a notation for the derivative of F or fix. Similarly, we can write the study with us D by the X off ffx. So this is another way off writing and everybody this d by the X is the derivative operator . Great. So this is the radio operator, so no, let's see ffx can be also routine s by Let's affect physical do way So this derivative can Britain s divided by NeeIix So this is also just another way operating a derivative If we can assume this F X is equal to buy right? So these were some rotations. So for writing are derivative basically mathematically right It's OK. So now let's take an example for calculating the derivative of a function with the help of our first principle. Okay, so let's see an example. Let's say our example is we have to compute the derivative off. Why is equal to F X is equal to signings by first principle. Okay, so now let's evaluate this. So what was our first principle? It was f Premix is equal Toe limit extends to zero F off X plus EJ minus F off X. Over H. Now remember, this limit should always exist in order for this derivative to exist. So this limits should exist, right to know what is F off x plus edge. So we will substitute explicit in this function so that we will be that will be signed off . Express edge minus sign off X and over each. So now we have one identity. That this sign off a plus b is equal to sign it. Cosby Less corset sign be So let's expand this signing sign off. Explicit. So that will be sine x koszics Gorsuch plus Koszics. Sign it. Minus sine x over at. Okay, so, no, let's try to take something out. Something common on really change in a better way to compute the elements. So when we see we take this minus sine x as effective, This is minus sign ICS one minus sausage. And this is Koszics signage late on. This all is over, right? So now this one minus corsage can be written as you sign Square. That's right. Oh, by the identity one minus cost. Two x is equal toe do science quakes so we can easily substitute that here. Great. So that will be do signed square etch by do over the edge less koszics Indoor signage by h So by the algebra off limits, we can write it more clearly. Ah, so by the algebra filaments, we can substitute limit in this also. And this also so we can Let's make this more clear here so we can write this as limit Etch approach is 20 minus sine x that this is to buy edge and we can write the says sign etch by two over. Excavate too Hold square. And this also gets more people exploit to hold squared. Yes, I am splitting the limits also by the algebra off limits So this will be limited Extends to zero cost sex into signage by H Okay, the sign it by edge limit limit s chance to zero. The san is wedge is equal Ruben So this part will be for sex, right? So this will be your six less limit. That's tends to zero Dubai edge my people I back at square by four names. Limit extends to zero. Sign it by do over. Expired too is also ones that this will become like If he valued the limit, this will become signed edge. Baidu over. That's why do hold square so this will equal. This will be equal to one and reveal your dilemma, and the final thing will be it will be Koszics 46 less limit so we can big help off the previous steps. This thing will be that's very true, Frank. And this thing will be one, so this will be limited. It's tends to zero. It's a great book, right? And now limit extends to zero as bright overload approach to zero since its approaches to zero. So it's quite well also approach to zero. So the final lines that is coming as Koszics. So that means F Prime X is coming as or six late Hence, this is the derivative off Ffx is equal Do signings that is Koszics by the first principle of limited. So this is our answer guys. One thing that I forgot in between that is minus sine X is also getting multiplayer over here. So that will be This thing will be multiplied by minus signings also and let me make it bigger. So in the final step, this thing culture gets multiplied by minus sine x. Okay, But since this part is getting zero, so are on someone be affected. Like so our answer is again a frame X is equal to Koszics, right? 15. Algebra of Derivatives: Hey, guys, welcome back. In this video we will be discussing about algebra off derivatives. Let's say there are two functions F x and Dfx and also resumed that narrative off. Good. The functions are defined, right? So let's define the rules off algebra from derivatives. So first of all, there is additional, right? So what is the addition rule? It is be by D X that is derivative of a function which is addition off two functions f off express Dfx. So the derivative off this whole function is equal. Do this some off, innovative off what the functions that is D by the ex FX last nearby the X G affects right So this is the addition. Now let's take an example. Let's see, we have a perfect is equal to x square and defects is equal to one plus X right So d by d x off ffx less dfx will be anybody X off x well plus one plus x and that will be cool too. The body X off the first function. Let's be by D X off the second function. So this will be two x less This is zero less one. So this will be one for the answer will be to express one So we can say that derivative of a function which is itself. Addition off two functions is addition off the elevated off the individual functions. Right? So this was our addition rule? No, let's similarly, we take our second rule, which is this subtraction? No. In this affection, we have two functions which are getting subtracted from each other. And we have to calculate did very well off the subtraction off the two functions. So we have B by the X off ffx minus g affects. So you have to calculate the daily we do off dysfunction. That will be cool too. New baby X off ffx minus debate the X off jfx so we can see derivative off difference off two functions. Is your friends off negative off? But the functions right? No, that's a good example. Let's say if X is equal, do signings and do you fix is equal. Do let's say excuse Children innovative off a for fix minus dfx will be delivered to off sine X minus X cubed. So this will be anybody except for sine X minus D by the X off X cube. So this will be your derivative off Sine X is equal. Do car six and everybody off. X cube is equal to three x squared. So don't worry about the blood off sign excess Koszics with the on. So we will be covering this in the diplomatic section. Separated or no, just remember that dedicated offside access Koszics. So this is our subtraction not totally as the broader cool. This product is actually little different from the addition and subtraction. It's not as simple as theoretician and subtractions rules we just discussed. So what is the product? The product As when we have to calculate the derivative of a function. It is my application off two functions. So let's say we have a function like the FX indoor GXE, right? So in this case, the derivative will be fast function. It will be first function multiplied by very, very give off second function, unless we can see a little bit of off, we can see it. They never do awful fuss function multiplied by second function. Right? Also, uh, we have some easier way to remember this. So there are poodles, basically, in the first time we have one function multiplied by the everywhere live off other function and the second element is opposite. That is the river do off. The first function multiplied by the second function, so we can simply write it as first function times. Second function very well. Let's first functioned very with you and second function where this dash or prime in honesty. So this is an easy way. You remember this room? No. Let's try to make this more clear with the help of an example. So let's say we have for fix is equal. Do Koszics and G X is equal to X. Let's see. Then we have to calculate Lieber the X off FX multiplied by GXE that will be called six multiplied bags. So how we will complete this study with you that will be first function that is cost six multiplied by the derivative off second function that is deep I d. X off x less first function Deliver that istea by D. X off course six multiplied by the second function that is X so that will be cost six multiplied by anybody X off X, which is one less day by the X, of course X this is minus sine x. This thing will also cover So expert now, just remember that will be minus annex in new X to the answer will be costs X minus X sine x So this was the product. Now let's take the last rule off the algebra off derivatives. That is the question truly now this is really difficult. Are ruled out of this out of the four rules. So we will cover this a big slowly. So the question rule is, let's say we have to calculate everywhere off that is deep addicts off a function which is actually a division off two functions. Let's say FX by GXE Now remember, this GX should not be equal to zero because once this G X is equal to zero, this thing will not be defined. Okay, So if this thing is true, then anybody x off effects by GX will be equal group. It will be equal to let's say this is as the numerator and this is the denominator. So let's The result will be the normal function that is GX multiplied by the derivative off the numerator function that is D by D. X off ffx minus numerator function might replied by derivative off the denominator function on this thing, holding is over denominated function wholly squared. So in short and tradition, we can write it as the nominator multiplied by enumerated everywhere, too, minus numerator function multiplied by the nominal derivative over the normal, it'll hold square, so this is a short envy in the form off notations. Remember this rule, right? So this is a short tradition down form of the question ruled easily. Remember this room? No. Let's take an example. We're F X is equal to X and G F X is equal to one plus X, right? So now let's go by the question to calculate the derivative. This will be the body itself x over one plus x. So this will be denominated function that is one plus x multiplied by derivative off numerator function that just d by the X off X minus numerator function that is X multiplied by D by d It's off the normal to function, and this whole thing is over a normal function. Hold square. So the answer comes as one plus X multiplied went now D by D. X off X is one so this will be one minus x. No, this d by the ex Open per sex Here we can apply the additional So everybody x off one person X will be d by D X off one. Does anybody expects to? Anybody except one will be zero plus t body X off X It just one. So this thing is over one plus X whole squared. Right? So the answer is one bless eggs minus. This is excellently and this ex gets canceled with this X. So it is 1/1 plus ex old square. So this is the answer. So, guys, these were the four rules off the algebra. Often derivatives. So this was it for this video? Hope you have understood the rules. See you in the next video. Nobody. 16. Derivative of common functions: Hey, guys, will you come back in this video? We will be covering the derivatives off some common functions, which would be very helpful going forward. Okay, so there will be a list of common functions for which we will be listing out the derivatives. Okay, so let's start with that. So our first is the elevator off express to the power and that is D by D. X off extra, the power. And so every time women mean very well, we will be mentioned it using divide the X. So currently we are differentiating with respect to X so dx, right? So we will be using the by the extreme place off derivatives. So D by the X off extra the bar and is end times X to the power and minus one where n belongs to the domain off real numbers. Okay, so this is our first rule. No. Second is ah very often used. That is D by the X off a is equal to zero. There is any constant function that this aloof is not changing with respect to X like if X is equal to one or Mexico do affects equal to three that is any constant number. So d by the X off a constant is or Macedo not the 3rd 1 is multiplication off above two functions. There is debate the X off a multiplied where any function that here is the constant and it fixes any function dependent on variable X. So now we can take here the constant out of the different station. This could be verified by using the product will off different station and hence it will become a multiplied ready buddy x off ffx. So we always take the constant out off the differentiation whenever it's in product with a function. Now the fourth rule is DVD X off. Say Nix is equal to Koszics. This we have already seen when we were calculating everywhere They're using the first principle after a video but we found that the body itself sine X physical Duke Koszics Similarly, you can prove the other day with those also like be body X. Ofcourse I next comes as minus signings. So you improved this also using first principle off elevated and I agreed you eyes to trade yourself approved these ISMs. So next one is D body itself Annex. This is a cool do sex clinics. And we know that psychics physical one by Koszics. Now, the next one is anybody X off Kosik de body itself. Course it kicks, which is reciprocal off sine X. So the debate takes off. Caustic X is equal to minus course. Psychics multiplied by cortex Airport access the reciprocal off panics. Now the answer is D body X off seconds, everybody, it's off. Sex is sex multiplied by annex. The next one is the body aches off cortex, which is equal to minus six Qualex. And the next rule is but the then through list D by the X off clinics. This is a cool do one by X on the 11th 1 is D by D X Uh, no. In here. The Ellen here is logged. This e right, this is Ellen X is equal to log base. E X is a natural base, which is Ah, generally it call to 2.71 approximately So normally, when we say log X, the basis 10 but in clinics, the basis e unease about 2.7 minute and so on. So this thing is readiness Ln X so d body aches off clinics is equal to one backs. OK, now the next one is anybody except for eight of the Bader X, which is equal to a to the power X Ames l any is any real number and it's great. It is greater than zero. So this condition should hold in order for the steady with to be defined Now, the last common function which will take is D by D. X off E to the power X e, as you know, is the natural base and its value is 2.7 minutes. So anybody except feet apart X is e to the power except self Onley. Dysfunction has its derivative exactly similar to itself. So derivative off Either the Rx remains the same as each of the parks. So these were some common function derivatives like the other, some common functions with derivatives. You should remember, as these comes very handy while solving any problems like we saw in the capital rule for evaluating elements. Okay, so let's take some examples based on these rules, let's start with our first example. So, guys, if you remember those rules which we just define that will be really helpful while solving problems related to these limits and elevators. So now, without any further delay, let's start with our first example. It just d by D. X off a to the power X multiplied by sine x So let's say this is our first function f x and this is our second function. G affects. Then this is multiplication, right? So we will be applying the product. We're here since two functions are in multiplication and ask for the product fuel we have first function multiplied by D by D. X off the second function plus de body X off. First function multiplied by second function. Okay, the knowledge substituted the values off effects if affects NGO Vicks So if X is a little bar X and G exist, Jeff Excess signings plus D by D. X off aided apart eggs multiplied. But dfx, we just signings here. So this is a to the power eggs and the brilliance of sine X is Cossacks from the previous rules, right less D by D. X off into the barracks from the rules mentioned earlier. This is also we can derive from there that it is it is a part ex times Melanie on multiplied by signings, which is here the second function. So the result comes becomes after taking a little bar X common. We get co sign X plus l in a turned up. This is any signings. Great. So here we have used two things. One is the product rule on the other is the river developed some common functions like sine x got got six and the dead will you off out of the box. So this was our first example. Now let's take a look at another example which is very we will be using the identities and the algebra of teddy bears. So let's see, our second example is something like we need to point the oh, any would do? Oh, that he would go off. And the next year I did by Annex. So we have to calculate the derivative off dysfunction so we can consider this function as ffx and this as geo fix. Where effect affects is the numerator function and dfx is the denominator function. So as we can see here that these two functions are in division. Okay, so we will be applying the question rule in this example. So now let's apply the question for. So as you might Askew guys might remember that the question rule is be by the ex hair debate the X off the next over an ex. The question will be denominator multiplied by debates off numerator That is D by D. X off clinics minus new Merida multiplied by anybody except denominator. So that is an X here, and this whole is divided by the denominator. Whole square, that is tonics, Whole square. Okay, No, this is annex multiplied. But the bodies of signings, which will you remember with you guys remember, from previous mentioned identities, that is one by X minus clinics, No delivery. Wolf Annex is sex clinics. Right? So this is six quicks on this hole is divided by an ex hold square. So this is our So these were some basically examples which were which were covering the application off the algebra of derivatives as well as the directives for some commonly used functions. So I hope you guys might have understood this lecture, Andi. So see you in the next video, where we will discuss about the General. Thank you. 17. The Chain rule: Hey, guys, welcome back. In this video we will be discussing about a very important rule for different station, which is called The change will use this rule when we need to difference. See it Ah, function, which is a composite function. Now let's say we have a function at to fix It is equal to F off g off X and this is a composite function by because the input perimeter off this function f is itself a function . Let's say if we call this as the inner function and this function will be our out of function. So the outer function is F or G off X, and the inner function is g affects and do you affects function is dependent on the perimeter X. So this is a composite function and chain rule comes into the picture when we need to differentiate a composite function. So let's right the general. What exactly is the change? Will the chain rule is for the differentiation off composite function and here we need to differentiate. It affects. So that would be debate the X off it affects no, in orderto achieve this. First we need to difference. See it this composite function with respect to the in her function, then multiplied by different station off the inner function that is G affects with respect to the perimeter on which it it is defined, which is X here. So in this case it is divided me x. So we will differentiate this G of X with respect to X and this defines the chain rule. So it might be looking a little tricky as for now, but this is actually very easy to memorize in order to remember this First, we need to differentiate this whole composite function with respect to the inner function that is G of X here. And after that it is multiplied by the differentiation off the near function with respect to the perimeter on which it is defined, which could be anything. And in here the inner function is defined on Perimeter X, and hence we differentiate with respect to X. Hence we write it as the body X off JFX, and this is the change will. So this is the very important role when we talk about derivatives. This will really help you in solving a vital idea of problems when we're really dealing with derivatives. So this is the change Will Now we will take some examples to make this concept more gear. So let's start with our first example. So the question is, do fine. Divide by D. X there. Why is equal to sign X Whole cube? This diva already X is nothing but you buy the X off a function. So this is the other way off saying D by the x of I as diva dicks. So these are equipment and the boat mean different station off I with respect to X and here bicycle sine x hold you right now as you can see it from here, this is a composite function. So let's define our inner function here. The inner function is geo fix is equal sine x and the outer function is f X is equal to x cubed. So when we do place expert GX in ffx then so it didn't outer function X cube, So it will become sine X whole group. So when we take X in place of G X in the upper function, we get excused and hence weaken right f d off x as sine x hold you No, we will apply the general here. Okay, so how do we write this as we'll write it as Diva Body X will be differentiation off the composite function with respect to the inner function, we just signings here. So that is D by D sine X. The Cenex meets differentiation differentiating the above function with respect to signings . So since it is the inner function and then we multiply it with different station off the inner function with respect to the perimeter, this in function is defined that this x so d by D x off signings. So as we have discussed it previously, anybody X off X cube is three x cure Similarly, we can see if you replace X with some other valuable It's a E and we have to help pull it. Give my GT off Peak Cube. Then again, it would have bean three key square. So in the similar grounds, we can see d by d x off sine x whole cube When the input parameter is sine X is three signs Square X are reconsidering sine X holds where both had the same thing multiplied by the difference Asian off sine X with respect to X so as we have seen previously, debate DX off sine X is Koszics. So the guys the final result for this composite function is coming as three signs Square X course or sex, right? So now let's take another example on General. So let's see, why is it called one plus X Square Whole square? And we need to find the vibe ready X For this, since this is also opposite function were the inner function G affects when this whole is Foggo fix, then the GX the inner function will be one plus X square and effects will be X square. So if you plug in GX inside this FX So if we replace expert the value off Dfx, then this will be G affects whole square and that will come out as one plus x square. Hold square. Right, So this is the inner function. And, uh, this is the outer function. So now let's apply. The change will so divide Body X will be the differentiation off this composite function that is 11 less x square, whole square vid respect toe the inner function that is one plus x squared, multiplied by the difference Asian off the inner function, which is one plus x square with respect. Oh, the perimeter off the inter function that this X So this is the chain route. Now, as we know, Defense Station off X Square with respect to X is two x So if we replace one plus x square with the also then different station off the school with respect to be will again be duty so and the similar. Brown's a different station off one plus x Square Whole square with respect to one plus X Square is two times one plus x squared. Okay, and now we find the difference Asian off the second part. So the difference Asian off one will be zero and different. Station off X square with respect to X will be two weeks, So this will be two times one plus X squared times. Who expected would week four times when Bless X Square. So this is the differentiation off this composite function as we found out by the help off general. So now let's take one more example on this gym room. So let's see, Let's take an example now, which is a little complicated. So now our example. Next example, is, let's say why is equal to X and the Knicks less three whole square. And this thing is holed square where Ellen X is logged this e x and we need to find the fever body aches in this example since this is also a composite function, So we will first find the inner function. So the inner function G affects will be this part that is Exelon X less three and the outer function F off X is if you freed this thing in x square, then we get our main function. That is therefore the affects. So the outer function f x will be equal Do X squared and hence we can Yeah, we verified this by replacing X ray GX in ffx. So then we get g affects holds where? So this will be g of X hold square. So this will be ex Elin eggs. That's three one square, just same as our mean complicit function which verifies that our assumption, foreigner function and our function both are correct. So every time we like NGX inside FX, we can easily verify are opposite function like this. Okay, So, as we have defined our inner function and the outer function? No, we can easily apply General. So the vibe i d x will be the difference Asian off the composite function that is X Ln X plus three whole square with respect to the inner function which is X Ln x less three multiply by the differentiation off the inner function with respect to the function on which it is defined all we conservatives with this part of the perimeter that this x so no as from our previous examples, we know that D by d. X off x squared transition of Xcor with respect to X is two x so similarly the different station off. This will be two times Exelon X plus three multiplied by the different station off Exelon X plus three with respect to X. Now this Exelon Next plus three This can be Britain s our first part of times Different station Different station off three is zero plus D by D. X off excellent x So we're big breaking by the algebra of derivatives So what remains is two times x Ln x plus three multiplied by D by D. X off ex clinics Now we need to calculate debate the X off X Ln X Okay, so let's write it So this will be two times x Ln X plus three on the second dome D by D. X off. Excellent x so we can use the product rule here to compute the d by the X off X Ln x it states that first function multiplied by different station off second function. Oh, so I guess, uh I wouldn't be body X before the whole thing. Yes, I have ridden the body X before, so let's just relight our dessert, you know? So here, first function times different station of second function plus second function multiplied by different station off extract will be one. So this gets canceled and our answer will be to times Exelon X plus three one plus Ln X. So the final result for the question is this So this was another example off, General, This was it for this video. Do join me in the next video where you see more examples by 18. Miscellaneous examples-Derivatives: Hi guys. In this video, we will be discussing some miscellaneous examples for the topics covered until now. So let's start with our first example. So the problem is to find very, really off with respect to X were vice given by nine X squared plus five eggs on this we need to achieve by using the first principle. So let's start with the solution. So if you guys could recall by the first principle derivative that is F frame eggs, which we can also write, as do you ever DX as we need to find the deliberative off. Why, with respect to eggs, it's can be read in X f x plus edge minus ffx or what EJ and this is whole under limit Etch approaches to zero. Nor does F off explicit. Let's say this way is F of X. Only so f of X plus edge will be nine games X plus edge, Whole square bless five times X plus edge. On this FX will be nine x Square plus five x on this thing as a whole list do. I didn't at Let's see if it can simplify this so it opens as nine X square plus at square less to exit. And this nine next square would have nine s common on The remaining thing is ah, if the remaining thing is five off five times explicit minus X. So this thing is divided by so excluded gets canceled with this X square and this ex gets cancer with this X So what remains after this is limit Etch approaches to zero. And now we see that in the numerator all the remaining comes have in common and since it is not equal to zero so we can cancel out one x from what? The numerator and denominator. So the remaining things are nine times X plus two X plus five. As the escort cancelled from both the numerator and the denominator so now began. Simply put at approaches to zero here on evaluate the limit. So when it approaches to zero, this part becomes this whole part becomes nine times two X, that is 18 X plus five to the answer. Is it in explicit wide? Also, you can verify the results by using some of the properties which we have been discussing throughout the course. So you will tell were like nine times the widow of Extra, which is two X So this thing becomes 18 X. Let's fight as they develop excess one. So we have even very fired the correctness off our answer. Okay, Now let's take another example in which we need to find divide by D X for vie revise equal to sign X minus Koszics. Okay, so long. Let's see how toe calculate didn't even live in this example. So if we suppose this function as ffx and dysfunction is gov ICS then derived by D X can beat it in us D by the ec self ffx minus geo fix. Now you must have guessed Be to rule off algebra to apply here. Yes, we need toe apply the subtraction will there So the by subtraction will weaken right? This thing is D by the X off ffx minus D by D. X off gov ICS. So this is the subjection which were applied. So this we have already covered in a previous realize. You guys must be knowing that. So now the body aches off effects that this debate goes off sine x minus. Debate takes off jfx that this debate takes off signings. Now I have mentioned all the basic rules. What the river tubes off Comment Pragmatic functions So D by D. X off signings from there s Koszics Andy. Body aches of koszics is minus sine x So this is minus minus sine x and two negatives off this Lee combined to former positive. So this will be Hussein Express signings. So the answer is course explosiveness to the derivative of this function or debate the X Do you ever needs for this function is course express signings. So that was our second example And it was based on the subtraction will from the algebra for derivatives. So now let's take a look at another example where we need to find divide by D. X, then why is equal to Dubai X plus one minus X squared over three by three x minus one. So this is our function by and we need to calculate give everybody Except So now, as we can see, let's assume dysfunction as ffx this one this function is fo fakes and this function iss geo fix You know, as we saw in the previous example, uh, that you ever dicks can be didn't as d by D X off. Why it? Bias ffx minus geo fix. Okay, so as we have already assumed FX NGO Fix So the from the subtraction rule off for algebra derivatives, we can write this s D by D. X off ffx minus debate The X off do you fix and D by D. X is off of excess different station of dysfunction and the basics of G affects is different Station off dysfunction now in both off dysfunction effects and dfx weaken further applied divisional for off the algebra of day workers. Okay, as both of these functions are in usually our division off two functions. So to calculate the derivative of these two functions, we require the question too. All right. So, firstly, we could calculate the d by d. X off ffx using the question rule. And after that we will calculate the ready except Joe fix Gonna Let's first calculate the bodies off ffx so d by d x off fx that I reverted back Question rule is denominated function that is X plus one multiplied by the river live off numerator function which is to and that everybody comes as zero minus numerator times Everybody off the denominator, which is But since and this all is you had already nominated hold square. So first part is zero and the second part of minus two. So this will become minus two over X plus one holds where so the differentiation off ffx is minus two over X plus one whole score. Similarly, we will evaluate Derivative of G of X respecto x So deeper ticks off jfx. As for the question rule, our new knitted and denominated here are X square and three X minus one, respectively. So we'll write it as denominator multiplied by defense Asian off numerator That is the real value of X square, which is two X minus numerator. My department, everybody both denominator. So they would go three X punishment that is three minus zero on this hole is divided by three X minus one minus one Hold square. So that's no Simplify it as this will be six X square minus two weeks minus fi X square and this whole is divided by three X minus one whole square. So this is three X square minus two X over three X minus one whole square. So now we have successfully calculated their deals for affects nd off X respectively. So now we just need to march the visuals who get the final answer So D vibrate gigs is our dealer. Body X is de body X off ffx that is minus two over X plus one whole square. So the first part is minus two over explodes on whole square minus nobody except defects, which we have already calculated as three X squared minus two x over three x minus one whole school. So this is our final dessert. So the developer DX or you consider every with respect to X is coming like this. So we used to rules of algebra that it's subtraction and division rule Wellard than they were before dysfunction. Okay, so this was our third example. Now, let's take one more example in which we need to calculate developer the X We're bicycle do on the route signings. Afraid so. How do we calculate the steady rhythm as we see that this way is a composite function. But we can suppose, uh, this as the inner function on name it G affects on the outer function as root effects. So we need name. It does FX. So This is the composite function this whole. So you tried toe Explain it in another manner. Let's say we have an input variable s ex and the function is sane. So when we pass Ekstrom dysfunction, the output comes as signings. So and when this output this signings is passed through another function which applies the route to the import We get under root signings as the airport. So this is our competent function. This is X the super sent G affects this from this function and output of this is Dfx and this is F off jfx. So this is the This one is the motor function and this one is thing of function. Our function is that affects an infection is defects. So let's write it here. Do you fix? The Jew fix is equal to signings and therefore fix is equal Do and the rules say nix and hence f off Geo fix is equal to under root cynics So do you ever Dicks will be d by d g extras The inner function according to change will off the difference here this revenge of the whole composite function with respect to the inner function multiplied by the different season off the inner function with respect to the input variable on which this inner function is different. That this ex to know this F off G of X is and ruled signings And do you fix is signings. We need to gather it be by the sine X off under new signings multiplied by D by D. X off signings that it's Koszics Look, let's see African calculate d baby signing off under roof sine x So let's assume we need to calculate DVDs off under Codex the Beach weekend, right? As De Buddy, it's off extra devoured Half This will be won by two multiplied by X to the power half minus one that is minus off. So it is a person when like to root X so derivative of Fruit X, with respect to X, is coming as one by two weeks. Similarly, we can see that diva on. Let's another deeper be way off under Goodbye Bill, come as on back order. If I so similarly d by the signings of On the Road Sign X will be won by Blue San, it's when my total cynics so our ended that it will be deep by DX dy sine X often route sign Next times Koszics So this thing is one by little Senate's So our answer will become that is one by total signings aims course six And here we used the general to evaluate adviser. Okay, so now let's take another example on general only as it is a bit tricky And this would be our last example. So in this example here, we need to calculate the anyway off by with respect toe exiled only we're by here is e this to the power X square plus five x plus three. So this is our function in question on me to calculate Do you ever dx right now as you can see here as well that function vie that this function way is a When was it function Okay where this is the inner function G affects on this is the outer function f X This is the art of function. So let's writing it separately. S G f X is equal to x squared plus five x plus three and the older function is f X is equal Do e to the Bar X and if you plug in G affects in this place off X. In ffx, we find f f g off X is eaters to the power X Square plus five X plus three. So now let's apply the same route to calculate the dedicated Okay, since we validated, affects and effects. So now again from the general lecture we know developer deals will be derivative off the composite function with respect to the inner function multiplied by the diverted off the inner function with respect to the input variable which is X. So now let's substitute devalues. So G x is X squared plus five express three So and f of g affects is eaters to the Power X squared plus five expressed three mostly black but de body itself GX, which is X squared plus five x plus three. So the second part this part is very simple, right? And, uh, the first part this part is a little trick here. So now let's suppose this thing is someone other variable by so debate diva off e this to the power vai, which we know is it's the power bi we can take some other available that's a dusty so D by dp off to the party will again be to the party so similarly de by the X squared plus five express three off ears to about X squared plus five x plus three will be either stop our X squared plus five x plus three. Okay, guys, my replied by the derivative off the second time you just wicks as different vision of excursions to X plus five as different station of five X is five and bless. Zero is different. Station off a constant is it'll? The answer is eaters to the power X Square plus five x plus three multiplied by to express five. So this is the answer. So these were some examples based on derivatives. I hope you guys must have understood the concepts and enjoyed watching the videos. Thank you guys.