Transcripts
1. Welcome to the course!: Welcome to my course. Let me introduce myself on my course about differentiation just in the natural. I am Daniel Chico. I'm a mechanical engineer, but my main focus is on teaching you and others online. I've been in an accident. Mathematics faculty in secondary school. I cannot say that I was the best, but I've experienced the best training available. This inspired me to help you. I don't want to let you struggle with mathematics. It is a fine topic, but unfortunately, it is not always done. Well, I have several years of teaching experience, and I can still put myself into your position as a student. So I think I'm the one who can help you. I designed this course for students having trouble with differentiation. I'm going to tell you the teary in a nutshell, as I don't want to concentrate on solving problems. Therefore, I show you lots of numerical examples as I believe that this is the better way off learning topic. But it is never enough to see just the solution. I would like you to get involved. You are going to get practice problems, this hand for you. These are going to have to really understand the topic by applying everything you have learned. So what is going to be in the course? A short introduction to derivatives Based on calculating the limits off functions? I'm going to show you how to differentiate according to definition and how to check whether function is differential or not. Then I show you the special derivative results based on which you can differentiate functions quickly and efficiently. You are going to see how to apply differentiation rules and how to use basic mathematical tricks for complicated differentiation problems. By this point, you are facing expected derivatives, but sometimes functions can be defined in a clear my ex connection for implicitly or paramount likley developed functions. The methods are slightly different. You are going to be able to do implicit differentiation and aromatic differentiation by the end of the scores. This means that you are going to be able to differentiate any kind of single barai above function. Then I'm show you some applications. The basic one is determining the tangent lines to pervs, given an expletive implicit or paramedic matter. Then comes the Taylor polynomial, which is a useful approximation to besides being good practice for differentiation. But end of the course, you are going to know everything that you should know about differentiating single barai above functions. Let's start differentiating.
2. Derivative: First of all, let me introduce the definition of derivatives. It is not necessary to know the definition as we usually calculate the derivatives by the half off some special derivatives. However, to get the full picture of the topic, let me show you how to differentiate by the half of the definition off derivatives. Actually, the derivative maze, the result off the process that you are about to do and the differentiation makes the process off. Calculating the derivative off something, make sure to use the proper vert. You can see how the derivative off F ax can be calculated at point X note. Often remarked, derivatives by an apple strove in physics. We use the apple strove if we differentiate with respect to anything but time. If we calculate a 10 derivative reminded differentiation by a dot above viable anyways, now I use one opposed trough. As this is a first derivative for higher order derivatives, we can use multiple oppose troves. What is the expression on the right? Inside it is a limit calculated around axe note. It can be calculated based on the rules off the limits of functions. If you are unsure about the calculation off limits. You can check out my cars on the topic. Based on that, you are surely going to understand the upcoming calculations. Just in a nutshell. We should calculate the value off the quotient by the value off X approaches the value off X note. By this definition, we can only calculate the derivative at a specific point, which is indicated by axe Note. The derivative is the slope off the FX function. At the given point, the graphical representation halves the understanding the black curve belongs to the FX function. The quotient that we are investigating is the slope off the blue line as we take a step with X minus external clanked the function value increases FX minus effects. Note. As the value of X goes to extend out, the Blue land gets closer and closer to the red line, which is parallel to this low but accident. In the limiting case, the blue line has the same slope as the red line. So let's get back to the formula. X note is a given point where we would like to know the derivative of the function. We must substitute accident into the FX function to get F accident. This is simple, then we have two known values. Also, we know the FX functions so we can Selves to do it into the formula. For example, F X could be XX crab or sine X. We can just put that into the former owner along with known value off axe note and FX note . Then we can calculate the limit to get the derivative. At this point, you can see that it is pretty complicated to use this definition as it can only give results in specific points. That is why, rather, use special derivatives to calculate. I'm going to show you a couple of problems on how to differentiate according to definition . But if you do not really need it, you can jump to the lectures when I show you the special limits with which you can calculate faster and more efficiently
3. Differentiation based on definition - 1. Example: Let's see how to calculate the derivative based on its definition. In the first example, we have to differentiate square root five x plus one at X Note. Exhausted tree So we are going to look for a derivative in one point without differentiating the whole function. Let's use the definition the limit off a caution must be calculated as X goes to X out. The difference off the function values in X and X note is divided by the difference off X not on X. We can use this form, but there is an other form in which began work easier. We can introduce a new perimeter. Age is the distance off X and X note so it can be calculated as X minus accident. Originally, acts approaches acts Note. If we use age, age goes to zero. Denominator becomes age on the FX function becomes f X note plus age as X equals two X note plus age. This is logical. X note is known It egg first tree. We can substitute it into the FX function to get F accident, so that is also unknown. Construct. That's substitute everything into the definition. The function is squared five X plus one. We must substitute X note plus age so three plus age in the first case and then we must substitute three. In the second case, the denominator remains age. We can do the basic calculations in the nominator. There is 16 under the second square, so we get for As a result, there is 16 plus five age under the first square world. That cannot be simply fact. Let's just examine the value of the limit. As EJ goes to zero, there is 16 under the Scrabble. So the route is for four. Minus four equals 20 So the nominator goes to zero. So does the denominator. As age goes to zero, this is a 0/0 type limit. This cannot be calculated directly. We must reformulate expression. There is a little trick that works every time. If ascribed appears in the expression, we can get real described in the nominator. To do that, we can take the nominator change the minus sign into plus sign and we can multiply with the expression that we've got. So we multiply by squared 16 plus five age plus four as we cannot change the value of the expression. We should also divide with the same term. In the nominator, we have a multiplication that looks like a minus B times a plus B. The result off that is always a squad minus B squad. You can check this by multiplying term biter. There is 16 plus five age minus 16 in denominator. In the denominator, we can just formally fried multiplication. We do not really have to deal with that. The most important thing is the nominator. Now only five age remains in the nominator as 16 minus 60 Nick, 1st 0 This is why did the reformulation in this way? Now there is aged both in the nominator on denominator, we can simply five it it as it doesn't equal to zero. It just approaches to zero. But it can never be Sierra. After the simplification, we haven't easy task. The nominator is five and the limit off the denominator can be checked. Five age ghost zero as age goes to zero that for the term under the square root goes to 16 . The result off taking the square root goes to four. So there is four plus four in the denominator as age ghost zero the solution goes to fight over. Eight. This is the derivative off the square root five explosive function at X, not exhausted tree In practice, this is the slope of the function at this point.
4. Differentiation based on definition - 2. Example: Let's see another example, We should differentiate. The given function at accident equals to five. Based on the definition off derivatives, the function is one over X minus tree. We can substitute into the definition on do some reform relations. The definition is known. Both forms off the definition appeared in the periods. Example. I'd rather work with the second form, so I substitute to death. Mom, we are going to check how much the function changes by the variable changes by age, where this change is a very small number. Let's substitute. Firstly, five plus age takes the roll off acts, and secondly, five takes the role of acts in the FX function. We can do the basic calculations. Five minus three equals due to their further is age plus two in the first denominator on two. In the second eliminator, the limit off one over H plus two minus one over to over age is investigated as age approaches zero, the denominator certainly goes to zero. In the nominator began 1/2 minus one over to as a limit. The difference also goes to zero, so there's a 0/0 type limit. We must use reform relations. If you have fractions, usually you can solve the problem by putting them together by finding the common denominator. The common denominator is two times H blast you to get this out of the original denominators. The first fraction is expanded by two, and the second for action is expanded by H plus two. That is why we get to minus h plus do into nominator off the new function. Now we can do simple calculations to minus two equals zero. So on Lee H remains in the nominator and the denominator becomes to age plus four. After the multiplication, we must deride minus age over to age plus for by age. This means that actually minus age is divided by age. So we get minus one over to H plus four in the and the limit of this can be calculated pretty easily. The nominator is minus one, and the denominator goes to four. As age approaches zero, the quotient goes to minus 1/4. This is the derivative of the given function at accidentally close to five
5. Differentiability: Let's talk about differential ability. When does derivative exist? When can we differentiate a function at a given point? You can imagine this graphically. The function is different, she able at a given point. If one tangent line can be drawn in the point, this means that the same tangent line shall be drawn. If we approach the point from the left hand side on the right and set for this, the function must be continuous at the given point. This is necessary, as this ensures that the tensions, like drawn from the two sides, go through the same X note. Vinyl point, if two different by values are attached to an excellent value, there is a jump in the function value We cannot have just want specific tangent line. There's a tear. Um, that states if a function is differential at a given point, then the function is continues. At that point, however, the connection doesn't work in the opposite direction. Even if a function is continuous, it might not be differential. Continuity is necessary but not sufficient. There is a limit in the definition of the derivative. This limit must exist. Other vice, The function cannot be deaf initiated, we can introduce one sided derivatives. Similarly, one sided limits limit from below means that the limit is investigated by X is smaller than X note and X is approaching accident. By calculating the derivative in this manner, we get the derivative from below. By calculating the limit from above, we get the derivative from above. In that case, X is greater than accent out. These months, sided derivatives must be equal. This means that the limit and therefore the derivative, really exists. Generally, we can investigate differential ability by checking continuity first and the von sided limits. After that, if the limiting the definition exists, the function is differential. The existence requires the one sided limits to be the same and also to be a finite number. Actually, a sufficient mathematical condition could be given more precisely. But basically this is what's required. If we go back to the graphical approach, we need the function to be continuous, so there cannot be a jump at a given point. Also, to ensure the existence off a limit the land cannot break, the slope cannot change immediately. This is not a very strict rule. The basic functions are usually differential everywhere
6. Differentiability - 1. Example: let me show you how to check the differential ability off, especially defined function. We must investigate whether the FX function could be differential at X note. By choosing the proper Alfa, the FX function is defined by two parts. Both of them are different, sociable by their own. Only the connection is questionable if X is greater than one. F X equals two awful times. X cubed advice. F ax equals two X squared plus do offer is our choice, but it is only vom perimeter. It might not be enough to ensure the fresh ability. First, continuity must be insured. It is a necessary condition for the existence off the derivative. If the function is not continues, there is no point investigating the derivative. Continuity means that the function value must be the same on the two sides off the investigated point. So now the function value must be the same. For X values, smaller and greater come bomb. If exp approaches, we can calculate the van sided limits off the function to check continuity. Be careful in case of substitution. On the left hand side of X, across the one X is smaller than one, so f X equals Do X squared, plus two so delimit from the left, Shabby, investigated by substituting X squared plus to the limit. Calculation is very easy. X goes to one so X squared also goes to one the ad to two days, so the limit becomes three on the right hand side off. One X is greater dumb bombs, so F X equals toe all four times X cubed. The result comes simply once again, X goes to one, and so this X cubed. Therefore, the product goes to offer. The two results must equal to each other to ensure continuity. This provides immigration for awful awful must equal to three advise. The function is not continuous and it can be differentiated. So from now on, offer requested tree on the check whether the limit in the definition off the derivative exists. In the next step, the one sided derivatives must be checked. If they boast exist, they are both finite and they are equal. The derivative exists at the given port. Let's start with the left sided derivative X approaches X note, but X is smaller than accident. The definition off the derivative is used. We can substitute everything that we know if X is smaller than one FX egg verse two packs scratch plus two. The function value at one is three as one squared plus two equals two tree. So the nominator is X squared, plus two minus tree or actually X squared minus one appears in the nominator. The denominator is X minus one as axe note. First of all, now we should determine the limits off X squared minus one over X minus well as X approaches, while both the nominator and the denominator go to zero. This is a 0/0 type limit, but it can easily be reformulated. Denominator can be factored X plus one times X minus one equals two X squared minus one. Now you can see that X minus one drops out. X does not equal to one so X minus one is not zero. We can simplify after the simplification. The limit off X plus one is two. This is the value off the derivative from below. Let's check the derivative from above Now we should substitute the other part of the function. F X equals 23 times X cubed as X is greater downfall F accidental is still three as three times one cubic Arrested tree X note is well, these are substituted. Then we can't. He arranged a quotient. The nominator is three times X cubed, minus one. You can see that both denominator and the denominator goes to zero. As X approaches, the limit calculation is troublesome. Fortunately, we can factor the nominator. X cubed minus one equals two X minus, one times X squared plus X plus small so X minus one drops out funds again. The derivative equals to the limit off three times X squared plus explosive, or, as X goes to want the expression in the brackets ghosted, Tree said. The limits is nine. The left sided derivative was do, and we have just calculated that the right sided derivative is nine. So the once said it. Everybody's are not equal. The function cannot be differentiated at X note, even though the function can be continuous because ensure that it is both continues on the slopes are the same from this two sides off one
7. Differentiability - 2. Example: Let's see another example. This time, we're going to check whether the function is differential at X equals zero or not. The function is defined by two parts, such as it equals two x cubed. If X is not greater than zero and it equals to actually the power of four. If X is greater than zero first, the continuity must be checked. If the function is not continuous, it cannot be differentiated. Actually, now we check if the function goes through a specific point or not. Then we're going to check if the slope of the factions continuous or not, these are the two South questions is the original. Is there a break in the function curve? The limit off the function is checked from the left and right hand side of zero. Let's start by assuming that X is smaller than zero, but it approaches zero. In this case, F acts equals two X cubed according to the definition off FX as X Ghost zero. So that's execute. The limit is zero in case of the limit from above, X is greater than zero. Therefore, F X equals two x to the power of four. This also goes to zero as X approaches zero So they're often sided. Limits are the same. The FX function is continuous at expect 1st 0 In the next step, divine sodded derivatives must be checked. They should also be finite and they should equal to each other. Let's see the derivative from below the definition off. The derivative is used for X values smaller than accent out. X note equals 20 So we investigate the case where X is smaller than zero. So f X was x cute. The function value at 00 So that appears in the nominator. We know this function value as we have already calculated. Eight months. We were checking the continuity. Basically, we investigate X cubed over X where X cannot be zero x just approaches zero we can simplify , So we need the limit off X scrap. It becomes zero at zero. This is the value off the left sided derivative. This actually means a horizontal tangent line at X equals zero. This is really the case for the X cubed function. We should also check the derivative from above this time X is greater than zero. Therefore, F X equals to exit the patter of four. This is the only difference compared to the previous quotient. Axe note is steer zero, and the function value is also zero. After a simplification, we can see that the limit of X cute should be calculated and it is zero. Therefore, the van sided limits are the same on day are both finite number just to clarify things. If we get infinite result, we cannot say that the two infinite results are the same. We need finite results and we need them to be the same. Now we do know that the derivative of the function exists at zero as the function proved to be continuous and the one sided derivatives equal to the same fate finite number at the investigative point.
8. Special derivatives: you can use special derivatives to solve complex problems. The upcoming results are really used with no for both differentiation and integration. I included a lot of results. Some might not be necessary for you in your university course, but all of them are useful. If you want to solve real life problems, including differentiation or integration, you can download the table off special derivatives along with the most important operation rules. They are all included in a pdf file. Let's go through the basic derivatives. On the left hand side, you can see the original affects function, and on the right hand side, you can see the derivative of the function for any X. The F X and F derivative X functions belong together. The first throw belongs to the constant number. The derivative off any constant number is zero. The second row belongs to the polling. Um, er functions. If you differentiate X to the power off something, the power decreases by one and the original father is used as a multiplier. You can substitute anything into the power and does not have to be integer. It can be any rial number. This rule is valid for them and through X, for example, equals two X to the power off one over M. The power decreases by one and X to the power off one over on minus one is multiplied by from over. The derivative off Sine X is CO Sanex the derivative Off co sign X is minus sign acts. Don't forget about the minus side. The derivative off Tangent X is actually not a basic derivative. It could be calculated based on the differentiation of belonging. Toa Roshan's as pungent X equals two sine X over co sign X Anyway, the result is one over close on excess crab. Similarly, the derivative off Goettingen X is minus one over sign excess crowd. The derivative off Sine X and cause an X are very important. The derivatives off the inverse trap triggered pneumatic functions are also important, but they are mainly important for integration. If you only have to be a bit differentiation, you do not have to memorize these except if they appear in your exams. The derivative off arc sine X is one over square root, one minus x. It's crab. The derivative off arc co sign acts is almost the same. The only difference is a minus. Sign The derivative off Arc Tangent X is 1/1 plus X squared, and the derivative off arc Cotton gin is minus one times that. This had sporting no meal and three dramatic functions. The exponential functions on the logarithms is important. The derivative off E to the power of Pax is itself where e is the island's number. Both E and A are constant numbers which raced the poverty containing X. The directive off A to the power of X is a to the power of X times natural logarithms A. The power doesn't change in case of exponential functions. It can only change for foreign no meals. The natural logarithms is the inverse off yard X. It's derivative is one over X. If we take logarithms off acts with the base off A, the derivative becomes one over X times natural. A great um A. Usually the base of the local rhythm is either e or town. In case of E, we have natural algorithm. The hyperbolic functions are made out of exponential terms, but they're def initiation. Rules are more similar to jiggle Matic functions. Hyperbolic functions might not appear in your University course, but they are important for engineers, so I include them into this differentiation course. The derivative off hyperbolic sign is hyperbole. Co signed on with Aversa, the derivative off hyperbolic Tangent X is one over hyperbole. Co sign X scrap the decorative off hyperbole. Contention X is minus one over hyperbolic sign exits. Crowd. The area functions are the inverse functions off the hyperbolic functions. Similarly to inverse Trigana Matic functions, these are frequently used for integration area. Hyperbolic sine X is the easiest. It's derivative is one over square old ex scratch plus mom. In other cases, you should check the domain off axe. The derivative off area hyperbolic Osan can only be calculated if X is greater than from. In that case, the result is one over square old excess crowd. Minor small other violence. There would be a negative number under the scrambled. That is why the derivative cannot be calculated if X is not greater downfall. The derivative off area High Purple of Tianjin and Co Tangent X are the same 1/1 minus X scrapped. Actually, the result is just formally the same area. Hyperbole If tensions can only be differentiated if the absolute value off X is smaller than bomb. An area hyperbole coat engine can only be differentiated if the absolute value of X is greater down. Mom, this is advanced stuff. I just wanted you to have every important special derivative. Mainly, you can concentrate on memorizing the pulling a meal, trigger automatic exponential and logarithmic carry bodies.
9. Fundamental operations: There are some fundamental operations with which you can calculate practically anything based on the special derivatives. F and G are differential functions. This should be proved before using the following rules. But the usual functions like polynomial street automatic functions, exponential functions are defensible functions. In this course, we are only going to work with functions that are unquestionably continuously differential . For those the following rules apply the derivative off the product off a really constant and the function equal to the product off the rial. Constant on the very vertical to function, the derivative off the some or difference off two functions equals to the some or difference off the derivatives of the functions. This is what we would expect in the race. However, the product and quotient rules are not so simple. The derivative off a product off two functions do not equal to the product of the derivatives. Instead of that, we should take the derivative off half, multiply it by G on. Then we should take the derivative of G, multiply it by half and then finally we some the product. This product rule is frequently used. Make sure to memorize it. The rule off there awaiting the fraction can be denied from the product rule. Still, it is better to memorize this portion through to the derivative. Off F over G can be calculated in a few steps. First, you should raise the denominator to the second power. Then comes the nominator. The derivative off F is multiplied by G and then the product off F and the derivative of G is subtracted. It is important that F is the orginal nominator and G is the original denominator. The roles cannot be switched. The fifth rule helps us to differentiate composite functions. The argument off F is an other function G X. In practice, you can think off sine X scrapped, for example, G X. The inner function is X scrapped. This is the anchorman off sign, which means the F function or you can think off e to the power. Of course, I X CO sign exes in our function. It is the argument off the outer function, the exponential. You are going to see examples. The derivative off a composite function equals to the productive derivative. Off each function off the composition, there can be more than two functions. In this connection. You are going to see examples. I'm going to show you how to apply these rules
10. Differentiation - 1. Example: Let's start with the basic example. A polynomial shall be differentiated. F ax equals to exit the power off. Five plus six x cubed minus five xx glad plus nine. We only need to apply the derivation rules off polynomial on the some. We can look for the derivatives off each term in the some, and then we can add the results. If you differentiate a Polina meal, you should decrease its power and multiply the result by the original, however, therefore, the derivative off extra power off five is five times X to the power of four. The power is decreased by phone and the original popover appeared as a multiplier. We do the same with six times X cute. The constant multiplier doesn't change the method. You can simply differentiate ex queued and multiply the result by six. The day relative off X cubed is three times X scrapped. This is money plant by six and the whole result is added to the periods the obtained five times extra power of four. The decorative off five times X scratch is five times two x. This term is subtracted from the previous ones as originally, five X squared was also subjected the decorative off 90 just as the derivative off any constant number is zero. We have reached the end off. The solution we might do. The multiplication is at this point, but that doesn't really matter. The most important part is that Polina meals are differentiated like this.
11. Differentiation - 2. Example: Let's see how to differentiate the product off. Two functions. We should determine the derivative off squared X times three X squared plus five X plus six . We could calculate the product and then differentiate a polynomial, but that's practice The use of the product rule. The derivative off age equals to the derivative off F. Times G, which is after evocative times G plus f times g derivative. According to the product rule, we can identify F and G on. Then we can calculate their derivatives separately. Let square root XB affects. Actually, dysfunction is a pulling a meal just like excess quad or execute. The only difference is that the exponent is not in toucher. Exponent is 1/2. Taking the an through means a division by AM in the exponent. It is better to ride the function in this form because we have a special derivative. In this form, the GX function is three X squared, plus five acts plus six. Let's differentiate. We can start by differentiating F X. It is a Bonino meal, so the exponent is decreased by phone and the result is more deployed by the original exponent. The new exponent is minus one over to Andi. Multiply the result by one over to this result actually equals to 1/2 times squared x. The mind of sand in the exponent means that the recipe broke Al off X to the power off. 1/2 is taken Now we have after bodies become multiply it by g x we can just go be the GX function. Then we can calculate the second product. Its first term is halfbacks. We can cope with that too. You could write square old acts or actually the power of one over to Finally, we need the derivative off G axe. It is a pony, no meal, so we can differentiate it. According to the Polina me a rule, the derivative off X squad is to axe. So the derivative of three times X square is six packs. The derivative of five X is five and the derivative off six is zero. This is how we get the result step by step. If you face the more complex problem you can use, the same mattered. Sometimes it happens to ride the separate results down and then you can put them together at the end
12. Differentiation - 3. Example: Let's see how to differentiate oceans the derivative off four to the power off X over cosign X shabby, calculated. This is clearly a question so we can use the respective differentiation rule. The derivative off age can be calculated as the derivative off half over G. To get that, we take the scratch off the denominator and we calculate difference of two products in the nominator. The derivative off the original nominator is multiplied by the original denominator and the product off the original nominator and the derivative of the original denominator is subtracted. Let's calculate step by step the denominator off the derivative is going to be co sign X scrap. This is simple in the nominator beneath derivatives, both for to the power of X and CO sign X are functions with well known derivatives. The derivative off four to the power of acts contents itself on the multiplier off natural logarithms, for in case of exponential functions, the function itself always appears on it is more deployment by the natural logarithms off the base of the exponential function. According to the portion through, we should not apply the derivative off F by G. So we copied the original denominator, which was co sign tax. Then F Times G derivative is subtracted. The original nominator can be copied, and then we should differentiate. Co sign X. It's derivative is also among the special derivatives. The derivative off course sign X is minus sign tax. Finally, we have reached the end of the differentiation. If you want, you can rearrange some terms, for example, that are two minus science, which can be substituted by a plus sign. But the main point is to be able to calculate the derivative. To do that, usually you only need the fundamental operations and the special derivatives.
13. Differentiation - 4. Example: Now let's see how to differentiate composite functions. By the help of the change, the derivative off natural logarithms tangent for X squared shall be calculated. This is quite a task, but at the end it is just a composite function that can be handled step by step. Let's see the compositions. Four x is a function by itself. It is the innermost function. If you step one outside of it, you are going to find tangent. For X Tangent is the second inner function. The tension is Whitney in a logarithms. This is the third step. Finally, logarithms is squared. This is the outer function. Altogether four functions are involved. Anyways, you can use the changeable step by step the derivative off the composite function it first the product off the derivatives off each function que x marks the inner function. Four x The derivative of that is the last time. In front of that you find age derivative, which is the derivative off tangent for axe, and you can go on like this. Let's just start differentiating and the matter gets clearer. I usually start with the outer function on go inner and inner into the composition. The outer function is taking the squared off natural logarithms. This is just a simple pulling a meal like X X crowd. You can forget what is scrapped. You can just think off excess Kratt their relative off death can be taken on. You can write the actual function instead of X So the deputy off xxx crab is two x The rule off X is taken by natural logarithms tangent for X so we can rent two times that into the result. This is the first term off the product. We can go on to calculate the next derivative. We can forget taking the square off the function and we are looking for the altar faction in case off natural logarithms intention for acts. Here. Taking the natural logarithms is the outer function. So we are going to differentiate that the derivative off natural logarithms X is one over X now, Dungeon four x takes the roll off X as that is the argument off natural logarithms. So the second term off the result is one over tangent four x From here on, we can forget natural logarithms to we should calculate the derivative off tangent for tax . The other function is tangent, so that is differentiated. The derivative off Tangent X is one over cosine X scrapped. The roll off X is taken by four x so that is going to appear in the argument off course sign the third term off. The result is one over Kazan four x cracked. Finally, we can't forget tangent to only four x remains. That is the inner function. This can easily be differentiated. The derivative before exe is for this is the last term off the result. Now we could rearrange the product, but anyways, we have a result. You can see the logic behind the change. You can always concentrate on one function that you can easily differentiate by going through the composition step by step. The calculations fairly easy. Just don't forget a knitter.
14. Differentiation - 5. Example: I'm going to show you a good little trick. In this lecture, we should differentiate son X to the power of Arc Tangent X. There is no such function among the special derivatives. We have excess crowd which has constant exponent, or we have to to the power of X that has constant base. But there is no such function where the base and the exponent are both functions because handle sine x squared. Or we could handle two to the power of our tangent X. But we can't handle this directly. We should rearrange the terms somehow and there is a trick to do just that whenever you have to deal with the function. But excellent is duality, meaning that you can't handle it. You can bring the exponent down to get a multiplier. Instead, you just have to use the loo. Great make identities. But there is no logarithms in this function. I don't care that's used low grade of any race, so we are going to have to introduce logarithms. We take the logarithms off sine X to the power of art Engine X and various the result into the expand off. In this way, we use a function and its inverse function after each other. It is like doing nothing to the function, but it helps us to reformulate the function. You can check each of the power off natural logarithms. X equals two X. Now that we have logarithms in after tax, we can use the logarithmic identities. If it proper appears inside the logarithms, we can bring it down as a multiplier. The natural logarithms off sine X to the power of Arc Tangent X equals toe arc tangent X times natural logarithms sine X just like natural logarithms X squared with equal to two times natural logarithms axe. Finally, we have a function that you can differentiate. We just need to apply the change room on the productive. The outer function is the E at function. If we step inside, that we find a product are pungent, extends natural logarithms sine X. Here, natural logarithms Xanax is also composite function for which we need to apply the changeable Let's just go step by step. The outer function is similar to e to the power of X. Let's just differentiate that the derivative of that is itself. We can copy the whole expression as the rule off excess taken by the whole experiment. This is going to be the first term of the result. Now we take one step further on deal with the exponent. As I mentioned, it is a product. So we need the product. First the derivative off our tangent X shabby. Calculated. This is going to be 1/1 plus x crab. This is a special derivative that you can find in the derivative table. This derivative shall be multiplied by the other term off the product. So we multiply by natural logarithms sign acts. This is the first part of the product. We need one more protect the first term off that is Arc Tangent X. This shabby multiplied by the derivative off natural logarithms sign acts But that is also a composite function. The other function is natural logarithms and the inner faction is signed. Let's deal with the natural logarithms. The derivative off natural logarithms X is one over X as we have signed X in argument. The result is one over sign tax. Then we go on to differentiate the inner faction side tax. The derivative off Sine X is co sign X, which appears as a multiplier, So the decorative off natural logarithms sine X became one over sine x Times co sign acts. This is multiplied by Arc Tangent acts to get the second term off the productive relative. So we have done with the experiment. We have to find a result. If you want, you can substitute the original form off F X instead of the reformulated form. This is just a formality. It has the reader to connect the results to the original expression. However, you do not necessarily NATO reform. If anything, you have the result you could substitute and the axe that you need to substitute.
15. Differentiation - 6. Example: Let's see a more complex problem. The derivative off logarithms hyperbolic assign X squared minus two X Cubed times arc signed eight x shabby. Determined the derivatives off The individual functions are in the differentiation table, so we just have to consent right on applying the proper rules first. That's just deal with the first term. It is a composite function. The inner function is hyperbole. Cosan X, this is raised to the second power and then the outer function is a low great um, fashion. We can differentiate a logarithmic functions with the given base. The derivative off logarithms X is one over X times natural logarithms five If the base off the Liberator Miss five The room off X is taken by hyperbolic Oh, Sonics scratch. So that appears in the solution. Now we should go inside the low buried um we need to differentiate hyperbolic assign exits Crowd. The counter function is raising to the second power so decorative off a polynomial is needed. The derivative off X squared is two backs hyperbole co sign waas acquired. So we have two times hyperbole consigned acts in the solution. Now we only have one mortar product. The derivative off hyperbolic assigned X that is the inner function, the derivative of hyperbolic Osan X is signed. Hyperbolic X. We tried this down and we have the derivative off the first term in the difference. Now we can go on to calculate the derivative off the 2nd 2 x cute times arc sine eight x This is a product, so the product rule is needed. Let's apply that first, we're going to need the derivative off to axe. Cute. The derivative off X cubed is three times X squared, so we need to times that Don't forget the minus sign in front of the brackets. The derivative off the product needs to be subtracted as a difference appeared in FX. Now we have the derivative of the first term in the product. It shall be multiplied by the second term. Off original, we can write arc sine eight x as a multiplier. Down this ridge, two roles to X cubed can be copied. This is going to be multiplied by the derivative off arc Signed eight x Arc Side eight X is a composite function. The counter function is arc sign on the inner function is eight packs the derivative of Arc Sine X is one over square owed one minus x scrapped. There is eight x in the place of acts, so we can write eight X scratch under described. This has been the outer function. It's derivative shall be multiplied by the derivative of the inner function. The derivative of eight X is eight. With this, we have the complete solution. The derivative off the product is within the square brackets. As usual, you could rearrange the solution, but I'm going to leave it like this.
16. Differentiation - 7. Example: Let's see another practice problem. The derivative off two to the power of natural logarithms sine X to the father of acts. Shabby calculated. It is a pretty ugly, uncomplicated function but lets the already it step by step. The hunter function is raising to give them power. We know the derivative off two to the power of acts. It is natural logarithms two times two to the power of acts. The base of the exponential terms goes into the argument of natural logarithms, and the exponential terms remains in the derivative. In case of F X, we can copy the function and multiply it by natural logarithms to this is the derivative off the outer function. The next step is debating the luxury. We could calculate its decorative, but we need to stop for a moment. We are not going to be able to handle sine X to the barber off X. We need to do some reform relations as it worst doing that before debating the logarithms. There is no such special derivative where both the base and excrement are functions. That is why we have to do the reformulation. We need to get X out of the exponent. There is a logarithmic identity that does just that. Now we have a look. Great. Um but if we didn't have it, we would need to take natural rhythm offside act the power of X and them It would raise it to the exponent of E. I have already shown you this before. Now we can use the logarithms. It is already there. The exponent off sine X can be used as a multiplier off natural logarithms sine X. Now we didn't have any trouble. Some function we just have a product that can be handled step by step by the half off the special derivatives. Let's calculate the derivative of the product in the exponent. This is the inner faction of FX. The directive off acts is Mom. This shall be multiplied by the second term off the original product. So we copy natural logarithms sign acts. Then we need the first term off. The origin are productive. Multiplied by the derivative of the second. We can copy acts and calculate the derivative off natural logarithms sign acts. That is a composite function. So we use the change. The counter function is natural algorithm, the inner function ISS sine X The derivative off natural logarithms X is one over x. The roll off X is taken by sine x So we have one over sine X. Then we need the derivative off the inner function The derivative off Sine X is co sign X So the derivative off natural longer than sand X becomes cause an X over sign acts. This is multiplied by X. With this, we have the derivative off the original exponent. So we have reached the end of differentiation. Always remember to use the logarithmic identity if there is a function, is the poverty that is also a function. This streak is useful everywhere. No, just during differentiation.
17. Differentiation - 8. Example: Let's see. Another example. The derivative off Fifth root Aria Tangent Hyperbolic X over San Ex, Cute to the power off. Three must be calculated. It is a composite function. The other function is taking the fifth truth on the function inside that is taking the quotient to the third power. This can be considered together as one exponent. We take the quotient to the third experiment and we take the fifth root at the same time. This is an exponent off 3/5 as taking the fifth root means a division off the ex felon by five. So the altar function is actually taking the quotient to the tea over 50 power. This is differentiated as a polynomial. As the exponent is constant, the exponent is decreased by phone, so we get extra the power off minus 2/5. This is multiplied by the original export as the crew shin in the brackets takes the role of Actually write that down with the proper excellent what entire we can go on to differentiate the inner faction, which is a quotient. Let's apply the proper rules generally, the derivative off fo Virgie's calculated as after evocative times G minus halftime. Street derivative over G Scrat G squared Miss sine X Cubed, squared. That's the easiest part. Then we need to differentiate. The original nominator are tangent Hyperbolic X. You can find its derivative in the differentiation able it is 1/1 minus access crab. This shall be multiplied by the original denominator. So we cook this sine x cute and we have the first product. Then comes the second product. The first term is simply the nominator off the original quotient. So we can copy aria Tangent hyperbolic X from the original nominator. This should be multiplied by the derivative off the original denominator sine X cubed. That is a composite function by itself. So we should use the change will to determine its derivative. The outer function is raising something to the third power. The derivative of X Q This three times excess grad, we just need to write sine x instead off X. Then comes the inner function off sine X cubed. We are going to differentiate sine x. It's derivative is co sign X. So we need to multiply three sine x squared by Kazan Axe The square brackets contained the derivative off sign experience. By knowing it, we have the derivative off the quotient. Therefore, we have the solution for the problem
18. Implicit differentiation: In case of some practical problems, you have to deal with implicit functions. So let me show you how to handle the derivative in such cases because the function implicit . If we can't express why with the help off X or X, with the help off by if why is a clear function off X like by equals to sign X, we have an expletive function definition. If the function cannot be expressed in such a way, we need another kind of description we are using on equation to describe the connection between X and Y. Formerly, you can always write that F X Y equals zero. Your ex is an independent, viable and by is a dependent variable. So why is the function off X, however, and exploited by ex function can be expressed? We only know that by choosing the value of X, we always get the value off by two. If you substitute and the acts into the F X Y equals zero recreation, the equation can only be two for Val defined by. But how can we differentiate by X if we do not know the by ex function? There's a way around the implicit function f X Y can be differentiated and then the derivative off Why can be found? We can use the usual rules to differentiate immigration. F X y equals 20 The derivative of zero is zero, and the derivative off F X Y is something the devotees off. Why is going to appear in the creation? So by rearranging the equation, we can get the derivative. There is an important addition that you must always consider. Why is differentiated as a composite function as it is the function off acts? But do I mean by that? Let's see an example Why squared shall be differentiated by X. The altar function is by squared and the inner function is via X. You can differentiate outer function that gives you two. Then you multiply it by the derivative off the inner function. We do not know why x explicitly. So we just write that the devotee of why X is the derivative off my ex. It appears in the creation and we hope to express it at the at. I'm going to show you some examples to help your understanding
19. Implicit differentiation - 1. Example: Let's see how to calculate the derivative, often implicitly given function by equals to sign X plus why we cannot express on exploited connection where vibe would equal to a function only containing X. But I can only be expressed by using both extend vibe. This is an implicit function. We should determine the derivative off. Why x anyways So we know that X is the independent variable and VAY is the function off X In this manner, we are calculating the derivative off Why, with respect to X become formally differentiated recreation by simply differentiating both the left and right outside the relative off the left hand side is my derivative. We do not know the value off why derivative? But we do know that why depends on X So it's derivative with respect to X is not zero but something we might be something formally by the usual by upper strove. Now we can concentrate on the right. Inside it is more interesting. We are diff initiating with respect to x both x on by depends on acts or so dysfunction is a composite function. As the sun off a some is taken. Therefore the counter function is signed and the inner function is X plus y. Let's start with the sign the derivative offside axis goes on tax. So the derivative off sign experts Why ISCO's on X plus? Why this should be multiplied by the derivative off the inner function. The derivative off X is bomb and the derivative off. Why is by derivative just like before? Now we have the former result off the implicit differentiation, but we do not actually need this. We need the derivative off by as that has an important meaning. For example, the slope off the implicit function is by derivative. Let's try to express that, as always, the derivative off. I appeared after the differentiation become bring every term containing by derivative to the left hand side and bring everything not containing right derivative to the right. Inside, we can find the multiplier off. Why derivative on the left hand side, and we can divide the creation with that, but we can only defied if one minus goes. An extra sleight is not zero. If it is zero, my derivative would be infinite. That would mean vertical tangent line. If we throw the diagram off the implicated function, let's put that aside. If the multiplier off why derivative is not zero, we can divide with it. Finally, we get the derivative off. Why? This is still not an expletive derivative, but we can calculate the derivative off by at any point that satisfies the implicit immigration. If you choose X, you can find one y value that belongs to death acts. This gives you a point with to court nights. If you substitute the to coordinate in this result, you get the slope off the function at the given point that summarized a matted first, the derivative off. The given implicit in creation is calculated then by there, but it was expressed from recreation. After that, the X and Y coordinates off a point can be substituted to find a slope off the implicit function.
20. Implicit differentiation - 2. Example: let me show you another example off. Implicit differentiation on implicit function is given by the equation. X times y squared equals too hyperbolic. Co sign X to the power of five plus by We cannot express why expectantly but we're looking for its derivative. We must follow the matter off differentiating implicit functions. Let's take the derivative off both sides with respective packs by is a function off X. We should keep this in mind. Therefore, Vice squared is a composite function. My ex could be sign axe or e to the power of X, which is don't know, but that function is Scrat. So there is a composite function. Let's start on the left hand side. We have a product X is multiplied by vice crowd and boast eggs on by our functions Off X, the product through it can be used. The decorative off X is vom. This is multiplied by Vice Kratt. This is the first product. The second product contains acts and the derivative off by scratch. Why squared is differentiated as a composite function. The outer function is taking the square off. Why the derivative off by squared by itself is too. Why. But Then comes the derivative off the inner function, which is just formally the derivative off by. This is what we are actually looking for at the end. Now we have the derivative of the left hand side, and we can deal with the right. Inside there is a composite function. The altar function is hyperbolic. Assign. Let's differentiate that First, the derivative off hyperbole, costs and X is hyperbolic sine x. The roll off X is taken by actually power of five plus y. So we put that into the argument. Then comes the inner function. Actually, the power of five process by the relative of X to the power of five is five times actually the patter of four. And the derivative off by is by Apple strove. Now we can express the derivative off. Why Let's go like the terms containing the derivative and put them on the left hand side. Every other term is on the right inside, so two x y minus sign hyperbolic as the power off. Five plus by times. Why derivative it first too hyperbolic sine X to the power off five plus by times five extra father of four minus spice crap we can divide by the multiplier off by derivative. If it is not zero for that case, get the derivative off. Why? If the denominator is zero, the diverting just wouldn't exist. It is practically infinite in that case. But for every other case we have the solution. You can always use this matter. However, there is an alternative solution that appears in some university courses. I prefer the solution matter that we have just used. But I'm going to show you the other censorship matter too. Let's get back to the starting point. We are looking for the derivative off by acts, but we do not know the exploited by ex function. The first step off the alternative mattered is to rearrange the equation. There should be zero on one side off the creations. Now I substrate the right and sad from the creation. But we could sub tried the left hand side. So that's works to it Doesn't matter what you do. We just need a new f X y function that EQ verse to function off X and Y and also equals 20 f Axel I equals zero gives us a curve on the X Y coordinate system that is the curve of the implicit function. And we can only find it by substituting X values off points one after the other. Now I have to show you something that is a bit of higher lover knowledge. We need partial derivatives. The topic of partial derivatives belongs to motive, a reliable capitalists. And now we rather concentrate on single, viable cockles. That is why I prefer the other matter. But I don't want to leave you in the dark. Your teacher wants to use this method in case of partial derivatives. We differentiate the function by choosing bomb Viable to differentiate with one we def initiated by from Mariah Bow. We keep all other variables constant. So now if I, def initiate with respect to X Y, is considered to be a constant, the derivative off a constant number is zero f x EPA strove means that I differentiate Vance with respect to X partially Let's do it step by step. The first term is X Times vice crowd here. Vice squad is considered to be a constant if we are doing partial derivations, so it is just a constant multiplier. The derivative off X is Vance said. The result is one times vice crap. Then comes hyperbole, co sign, actually the power of five plus y, where y is a constant. The function is a composite function. The derivative of the altar function is san hyperbolic. After the power off five plus lie and the derivative of inner function is five times X to the power of four. The derivative off by is zero, as it is now considered to be a constant. Then F by Apple strove is calculated, which is the derivative off F with respect to buy when we calculate this X is considered to be a constant, and why is the only varietal by only depends on vie, and it doesn't depend on X in this case, if we do partial derivatives by this means, the derivative off extends rice crowd is X Times to buy. As the derivative off. Weiss credits to buy and axes just a constant multiplier. The derivative off hyperbole co sign is similar to the previous one. The derivative off the altar function gives us hyperbolic sign actually power off five plus my and the derivative of the inner function gives us bomb after the father of five is a constant. So it's derivative is zero. If you differentiate with respect, right partially the derivatives off my gives us the bomb. Anyways, we have got to partial derivatives. Don't worry. If this matter is not very clear, you can stick with the other matters. If you're still with me, let's calculate derivative off right With respect backs, it equals the miners. F x Apple strove over FBI EPO strove This equation can be proved mathematically. Now I'm not going into the details of that. You can simply substitute the partial derivatives to get the result. Why is it good for us? Become really fast And also it is easier to calculate pressure derivatives than impotent derivatives. However, the terroristic a background is more advanced. We have got the same result as before, so the matters are working in the same manner. If you are learning this topic in university or college, just choose the matter that your teacher prefers other vice. I suggest using the first mattered. It would distract you so much. Usually, why is the function off acts? The partial derivatives are just massing with this conceptions, So I rather stick with the first matter
21. Differentiation of parametric functions: in this lecture on going to show you how to differentiate Parametric Functions First of four. What aromatic functions are off course there is a perimeter. This perimeter is used to describe the function more precisely. The coordinates are given by a perimeter instead of having a by ex connection, X and Y are boost given by tea. If you want to plug the function, you should substitute different devalues into the function off banks and vie for a given T . You get one point on the diagram. Usually an X value off a function determines the value of the function, so there is one function while you for every X for expletive functions. This is the case in case of parametric functions. More vie values can be connected with the given X value. This is advantages. If you want to describe dramatic curves, let's see the paramedic representation often Talibs, for example. He is the perimeter of the function, which can change between zero and two by this is the domain. There is usually given domain for the perimeter, but that is true for every function. In case off by ex functions, X is usually between minus and plus infinity. That is why I don't indicate the domain and the race for the values off T between zero and two by this to coordinate functions described al IPs, the X coordinate is eight times CO Santee and the vie coordinate is be times scientist. Depending on the values off A and B, they are the length off the seven major and semi minor axes. The greater von ist seven major access. The smaller is the semi minor axis. This is an ellipse with the center in the origin and access better at X can't buy. If you would like to describe al IPs with explicit functions, you should at least use do separate functions. This is why this approach is advantageous. But you can use paramedic representations for any kind of dramatic objects or any kind of functions. You can just think off the s time as the coordinates are given as a functions off time. The coordinate functions described the movement along given curve in the plane. So how can we differentiate dysfunction? We are looking for the derivative off by with respect to X. When I say that we are looking for the derivative, I always think off this derivative. But there are alternative derivatives to like the case of partial derivatives. Or we can differentiate with respect to the perimeter. That is what we use now The derivative off by with respective acts equals to the quotient off to parametric derivatives. If you differentiate by as a function of t so you differentiate variety with respect to t on. Do you differentiate X'd with respective city and calculate the quotient off these results ? You get the derivative off by with respect to X, they look at the former definition of the derivatives. If you would simply try by DT, you would get the right over DX. Let me clear this. You cannot actually simplify de vie over DT because it is a symbol on not an actual quotient. I just said this simplification because it helps you to remember this connection. Basically, instead of calculating the viable the X directly, you can calculate dramatic derivatives and calculate the similar quotient. Just form Orting. When you see my efforts drove, it is usually the day, but it it expect to x or in physics, the derivative with respect to lacked. If you see why that it is usually the derivative with respect to the perimeter or in physics, the derivative with respect to time. This is, But you need to know in theory on, let's see how to calculate in practice.
22. Differentiation of parametric functions - 1. Example: Let's see how to differentiate aromatic functions. A curve is given by its parametric representation. The X coordinate is t to the power of four on the vile coordinate is natural. Oh, great. Empty The domain is not given. We only know that t is greater than zero as it is the argument off natural logarithms. Anyways, we just want to calculate the derivative off by X with respect to X at any point, I have shown you the general formula for this we are going to need the derivative off X and why with respect to city. And then we can calculate the derivative off by with respect to X as viaduct over extract. Let's find the derivatives with respective T X that eat first of four times the huge according to the rule of polynomial differentiation is the variable which is on the forced . However, the exponent is decreased by one and the original exponent is uses a multiplier by that equals to one over tea as the derivative off natural logarithms eggs would be fun over x. So we have two paramedic derivatives like this. Now we can calculate that quotient the negative off. Why? With respect to X equals two. Why that over eggs that this means one over D over 40. Cute. The result is 1/40 to the power off. Four. This is only valid if he doesn't equal to zero. Beth T can be zero anyways, as natural yogurt empty requires me to be positive. So for positive T values, this is the slope of the curve. And actually we have got it pretty easily. As I mentioned earlier, any kind of function can be represented with para meters. This functions might also be expressed in implicit or explicit form. This one can easily be expressed in accident. It for y T equals two natural logarithms de so a simple T appears in the argument we can express the by X from the first immigration the equals to force through X. This Camry substituted into the Vita e gration, therefore by X equals two natural logarithms forced route acts. You could differentiate the function in this form to you can check that the result is the same as before. Usually, we are not going to expressed explicit function from the aromatic representation as it is not necessary. But sometimes you could express the by ex connection
23. Differentiation of parametric functions - 2. Example: Let's see another example of differentiating aromatic functions. The Parametric representation often Talibs is given X equals to eight times Co. Santee and y equals two b times Society where A and B are constants. The narrative off Why shall be calculated with respect to X. As usual, we are going to calculate the Parametric derivatives on. Then we calculate that portions to get device over the axe. Let's differentiate angsty and variety with respect to the derivative off society is minus society, therefore acts that equals two minus eight times Scientist The derivative off sign T is co sanity, therefore right that equals two b Times CO Santee. These are fairly simple calculations. Now we can substitute into the general formula. My apple strove equals Dubai that over X that if scientist E is zero, the potion can be calculated. But in other cases we have the result. Actually, the derivative would become infinite if Santee is zero, this refers to a vertical tangent line for those points. If you think often ellipse, you can see that there are two points about the tangent line is really vertical. You can also use the co tension faction to describe the result. Based on the graph of the contention function, it might be even more easier toe recognize that the function goes to infinity at some points now this approach has been very good for us. The Ellipse couldn't be described by fun expected function, but we can really easily describe it by the half off aromatic representation.
24. Polar coordinates: the most famous and most important dramatically presentation is based on polar coordinates . You must know this parametric approach. Obviously, everything you know about aromatic functions is valid for polar aromatic functions, as it is just a special, kind off aromatic representation. This means that you already know how to differentiate in course of polar coordinates. But let's see what polar coordinates are and how to describe a curve. By their help, we're going to use an angle as Param Eter the X coordinate, seek first to our Times Co sign Phi and the Vine coordinate equals two R times sine phi where are is also function off five. That's take a look at the coordinate system. Usually, the points off the coordinate system are described by the heart off the axe and by coordinates. This is the description that the Cartesian coordinate system uses, but there are other kinds off descriptions. If you describe a plane, you need to coordinate. If you describe space, you need three coordinates, but you can choose any kind of coordinate. Now we are using our and five s coordinates. These are the polar coordinates given value off. Our and five describe the location off any point in the plane. R is the distance from the origin and five is the angle measured from the axe access. Fine. It's positive if you measure it counterclockwise in the classical approach. Why is the function off acts in case of polar coordinates? Our is the function off five. If we describe a curve, the connection between the Cartesian and polar coordinates are quite clear. Based on trigonometry, appoint is located at our distance from the origin so we can throw the red line to the point. The angle off the red line and the X axis is five. If you couldn't the X or via access with this point, there will be a right triangle. The red line is the hyper to news, and we can calculate the length off the side next to the angle and the length of the side opposed to the anger, the length off the side. Next, the angle equals to the X coordinate of the given point there. Four X equals two R time school sci fi. The length off the other side equals two divi coordinates, so Y equals two R times. Sci fi If you are unsure about definitions. Just use this throwing X and Y are always connected to are on fire in the same way the connection between our and five is but describes a specific curve in practice. The are five function is given, and that is where we can start a calculations. So why is this approach good, or is it good? It's quite similar to the cat is er system, but it can describe pervs that are hard to describe by X and by in case off circles or ellipses. It is better to use polar coordinates, however, in case of a line going through the origin we can't even use the air fire function to describe. The point of the line was coordinate. Systems have their ups and downs. In practical appliances, you can choose the more suitable on
25. Differentiation in case of polar coordinates - 1. Example: Let's see how to differentiate. If a curve is given by polar coordinates, the curve is given by color. Coordinates are as a function off five equals to five. Actually, there is no dependence on fire, but there could be we should determine the derivative off via fax. To do that, we must introduce X and Y based on the connection off the Cartesian and polar coordinates. X equals two R Times Co. Signed fine and by equals toe Are times sci fi. This is true for all cases. Now we can substitute the function given for our there. Four. X equals 25 coarse and fine and why he goes to five sci fi. This is a usual Parametric description off X and Y find takes the roll off t, but basically we can use the usual mattered. Actually, you can identify what do we work it? It is a circle with the center in the origin, and we also know that the radius is five. As R is the distance from the origin and it is constant, we can easily realize that this is a circle. This is just some additional fact, but let's go back to differentiating we have a parametric curve on. We're looking for the derivative off. Why it respect to acts as usual by Apple strove equals Do I Don't over extract here of I Don't is the derivative off. Why, with respect to find an expert is the derivative off accident race back to five. Let's calculate down the derivative off cosign fires minus sine phi. So extent equals two minus 575 The derivative off sine Phi isco sci fi therefore by that equals to five. Co sign five basic derivatives. Now we can take that quotient the derivative off by with Respect to X Equals to five Corps sci fi over minus five sci fi or simply consigned five over. Signed five to minus sign. This has been very simple problem, but you can see the matter. This is how you can combat for our coordinates into problematical given cartage and coordinate on it is how you can calculate the derivative in case off such aromatic coordinates.
26. Differentiation in case of polar coordinates - 2. Example: Let's see another example. A curve is given by the air fire egotistical sine phi function, and we know that our and five Arctic polar coordinates we should calculate the slope of the curve in the X Y coordinate system. So we're looking for the derivative off. Why X? Let's introduce X and by X equals two R times. Cosine, Phi and Phi equals two R times. Sci fi as booths are and FYI are only depending on five X and Y are also functions off. 55 is the perimeter now did are five. Function can be substituted. Accident first to co sign Phi Squared by equals two. Goes on five times sci fi This is the Parametric description off the coordinates. From now on, we can calculate the derivative based on the matter off aromatic differentiation. Divi over the ax equals two. Why that over extent whereby that and eggs not at the derivatives off Y and X with respect to the perimeter five. We have to differentiate ex fine on by five. AEX Phi is a composite function. Co sign is squared. That is the outer function. The derivative off X squared is two x so similarly the derivative, of course, and scratch is to times co sign. This is more deployed by the derivative off the inner function. The derivative of co sign five is minus sine phi. Therefore, eggs that egg versus the miners to Times Co signed five times sci fi. The other function is a product off course sign and sign, so the productor with must be applied. The derivative off co sign fine is minus sine phi. This is multiplied by the other term off the product. This gives us minus sign five times. Sign five. Then we have consigned five times the derivative off sine Phi, the derivative of sci fi isco san fi. So we get goes on PFI scrapped. We can divide by that by eggs that to get the derivative off. Why, With respect to X, the result is Sign Phi Squared minus co sign Phi squared over to Kazan five. Sci fi. By substituting any value for fine, we get the slope of the curve described by the original polar regulation
27. Tangent lines: the derivative off a function is the slope of the function at any point. Therefore, calculating the recreation off the tangent line is an obvious application. Let's see how to determine the tangent line. First, a function is given. This function can be given in several ways and your family irv it all of the possible approaches. The function can be given an expletive manner. In that case, why is expressed as a function off X, for example? Why equals to co sign acts? This is the simplest case. The function could also be given in implicated manner. In that case, we might not be able to express right as a function of packs. Rather, we have a new creation. The X by solutions off the creation mean the points off the given half. Eating with such function is more complicated than dealing with expected functions then that are problematic. Representations for that coordinates or other kind of aromatic approaches can appear. It is not particulary hard to handle aromatic problems. The derivative can be calculated as we have learned it. We are looking for the tangent line at X note. So we have a function given in any way possible and the point is also given. But we would like to know the line tangent to the curve described by the function. This point can be given in several ways. Be accidental, Dwyane Out is the investigating point In case of exploited functions, Usually the axe knelt coordinate is even and then we can calculate vie note in case off implicit functions. Both of the coordinates must be given in the Parametric case. Usually the value of the perimeter is given at P. We can substitute that perimeter value into the Parametric EQ rations and we get the coordinates off the point. So somehow the coordinates off the point is now the equation off the tangent line is my X equals to my ex not plus my EPO strove accident Times X minus tax note here X note on my ex note are the coordinates of the investigated point. These are known constant. The derivative off by can be calculated with respect to axe than the derivative can be calculated at the given point by substitution X Is the independent viable? It can be changed as we go along the tangent line by is the dependent variable to the value of X litter minds the value off by Let me tell you how to memorize desecration. The tension Klein touches the curve at B. Therefore, the vine coordinate of the tangent line is my ex note at that point, the slope of the function and the slope of detentions lines at the same at P. Off course, The slope of the tangent line is always the same as it is a line. So we have a line that goes through B and haven't given my poster off Accel slow to get a line with a give us look. You should multiply the derivative by axe, Then, to ensure that that line goes to be the second term becomes zero at axe. Note. So why dig first? Why axe? Note. If X equals two packs, not there are other forms in which you can see this immigration divided relative equals to buy X, managed by accident over X minus accident form is also quite expressing. This is similar to how the derivative is defined. Just try to memorize one of the forms and use that. As you can see, these are just for money difference
28. Tangent lines in case of explicit functions - 1. Example: let me show you how to determine the tangent line. In case of an explicit function, we should determine the tangent line to the curve given by why X at X existed to the equation. Off the curve is my X equals two X minus four over X to the power off four minus eight. Cute. The generally creation off the tangent line is now X and y are. Mariah builds the other terms at the perimeters that we need to know. Axe note is given by Ex Knelt on the very body that axonal Chappy calculated Divi X function is given in explicit form. So why X note can easily be calculated. We can substitute to into decoration. The calculations can be done. The result is, miners do over 512. This is the second coordinate of the point about the tangent line touches the given curve. Now we only need the derivative at that point to calculate the derivative there. Let's just differentiate the function based on the differentiation rules. You could also use the definition off derivatives, but that's a slower approach. A caution must be differentiated. So the potion throughem shabby, applied first, the denominator can be scrapped, then we need the derivative off the nominator. The derivative off X minus four is one minus zero. This is multiplied by the original denominator after the father of four minus eight cubed. Then we subtract the product off the original nominator and the derivative of the denominator. The Denver native of the denominator is a composite function, so you should work carefully or you could just expand the term by actually calculating the cube of the faction. I prefer working with a composite function. So let's consider the third power ver as an after function and the poor in a meal as an inner function. The derivative off excuse this tree X squared. So similarly, we get three times after the power off four minus eight scrap. This is multiplied by the inner functions derivative we get for X cubed after differentiating extra power off. Four on. We get zero after differentiating minus eight. Now we have the derivative. We can substitute the X coordinate of the investigated point to find the slope of the tangent line. X note. Eggs Worsted to is substituted the operations camp. If you don't have a calculator, you can simplify by the scratch off to the power of four minus eight. After that, it is easier to calculate the result. If you have a calculator, you can just substitute everything. After the calculations and simplification, we get 25 over 512. As a result, this is the slope of the curve at the given point and this look off the touch of life. We know the three para meters that appear in the creation off the tangent line. Let's substitute down the equation. Off the tangent line is y equals two minus two over 512 plus 25 over 512 times X minus do.
29. Differentiation in case of polar coordinates - 2. Example: Let's see another example. A curve is given by the F X equals two X squared plus B to the power of two X function. The tangent line to discover of shall be calculated at X equals two natural logarithms to the general form off the recreation is known. X note is given my ex note invite their better accident, Charlie calculated. Don't get disturbed by the FX function if an expletive function is given. This F ax always equals to buy X when you want to put F ax X is on the horizontal axis on the values off F acts are on the vertical axis that access belongs to buy. So if we want to get my ex note, we can just substitute X note into F ax formally beacon easily substitute. We can calculate the value of the result by a calculator, or we can just reformulate to have a shorter result. It is also good for practice. If you wanted to reformulate, it would be nice to get natural logarithms and E to meet two times natural logarithms. Two equals two natural longer than to scrap, or you can even use each of the power of natural longer than to scrap. This means two squared. So why accident Egg 1st 2 times natural logarithms to plus four Or really, you can just calculate this by a calculator. If you are satisfied with on a proximate result, we still need the derivative of the function and given point. Let's differentiate f acts. It is a some so we can calculate term biter. The derivative off exits crowd is to axe E to the power of To Wax is a composite function. Let's start by the outer function. The derivative off the exponential terms is itself. So we have eaten the father of two acts. This is more declined by the derivative off to acts. So too, is them with supplier. This is the derivative, and now we can substitute natural logarithms to in the place off acts. Now we have the required derivative. We can use the calculator to get a result, or we can just reformulate similar later. The previous reformulation This look off the tangent line is to natural logarithms to plus eight. We have every necessary later X note was given. My ex note was hoping by substituting X note into FX on my derivative axe note has just been calculated. The Equation of the tangent Lankans, Britain, as usual, my exhausted to natural logarithms to plus four plus to natural longer than two plus eight times X minus natural logarithms to
30. Tangent lines in case of implicit functions - 1. Example: in this lecture, I'm going to show you how to determine the recreation off a tangent line. If the function is given an implicit manner, the function is given by an implicit equation. X times natural logarithms fine. Plus why times natural logarithms X equals to bomb. We're looking for the tensions line at b Note the coordinates off the points are one, and if you substitute one for X and E for by the equation is full feared. This means that the point is really on the curve, given by the creation you already know the generally creation of tangent lines. We can use this immigration for any case. The main difficulty is to determine the derivative off why, in the implicated case, the other two perimeters are already given. X Note is the first coordinator of keynote on vinyl is the second coordinate off being out so they are one and e respectively, or only task is to differentiate the creation implicitly on substitute the coordinates of investigated point into the derivative. Let's differentiate both sides of the creation with respect to packs. Why is a function off X? We differentiate implicitly. Keep this in mind. The derivative off the right hand side comes pretty easily. One is a constant. It's derivative is zero. The left hand side is more complex. There are two productive for which the productive must be used. Let's start with X times. Natural logarithms by both X and natural Logarithms are functions off acts. We really need the product. Herbal. First we need to differentiate X It's derivative. This bomb This is multiplied by the other term of the original product. Therefore, we have one times natural logarithms. By then, we haven't other product X is the first term off the original product. It is multiplied by the derivative of the second. Don't forget that by is the function of acts so natural logarithms Why is a composite function? The derivative off natural logarithms is won over by this is the derivative off the outer function white. Everybody is the derivative off the inner function. We can write this as we do not know more about the explicit connection between accent by this quad brackets now contained the derivative of the first product. That's differentiate the second product to the steps are the same. The derivative off. Why is my first strove this is multiplied by natural logarithms X. Then we have right times the derivative of the second term. Off the original product, the derivative off natural logarithms X is one over X. We have just differentiated the creation implicitly. Now we would like to calculate the derivative off why we can rearrange the creation to be able to determine the derivative off. Why there is x over y plus natural longer than x times. Why derivative? And there are terms not related to the derivative natural local vai vai over X camera grow to the right inside of the creation and then we can divide the whole creation by the multiplier off my derivative if it doesn't equal to zero. Finally, we expressed by derivative as minus natural logarithms via plus y over x over X alert vipers Natural logarithms max. To get this local investigated curve at a given point, we must substitute the coordinates of the given point into this creation. The X coordinate is one on the y. Coordinate is e Therefore, the derivative off by at X note equals the miners natural logarithms in plus E over one over e plus national longer than natural logarithms e close to one as he is the brace off natural algorithm that your logarithms bon equals 20 as one equals to eat a power of zero. These are special values by simplification the slope of the curve at being held its first minus e minus. T Scrat. By knowing this, we can find the determined recreation off the tangent Climb Hold off the perimeters off the generally creation unknown. So we can just substitute the equation off the tangent line off the implicit girl has bean out equals two e minus. E plus is granted times x minus one.
31. Tangent lines in case of implicit functions - 2. Example: let me show you another example of determining tangent lines to implicit curves. The investigated implicit function is given as a new creation. It is the power off X times y equals two. Why cute? We must determine the line tangent to discover that being out, we are looking for the tangent line in the usual form. As the tangent line goes through. Keynote, we can already substitute its coordinates. X knocked equals 20 on violent equals to one as these are the coordinates of being. The only perimeter that we need is by the river T at accident. Let's differentiate the function to find the derivative at the given port, we can formally differentiate both sides of the equation with respect to acts. Let's start on the left hand side. Each of the power off X times y is a composite function. The other function is exponential bomb. The inner function is the product in the expert. There were two of the exponential function is itself. So we have e to the power of X times lie in the result this shabby multiplied by the derivative of X times y as it is a product we must apply the product. That's where, according to that first derivative off X is needed, it is bomb. This is multiplied by the second term off the product. So we have one times. By then we ride the first term off the product down This is X which is multiplied by the derivative off. Why we can just write why they're putting formally altogether. We have the derivative of the left hand side. Let's continue on the right where we have a composite function. The derivative off y cubed is three times why scratch? But this is just the derivative off the outer function by is a function of X by itself that is in our function. So we have by derivative on the right hand side too. Now we can rearrange the extra gration to express my derivative. I brought the terms containing by derivative to the right outside we can divide by the coefficient off white derivative. If it is not zero, it is not going to be zero for the given point. But the solution is only correct. If we do make sure of that, we're going to substitute. Anyways, the coordinates off being out are zero and from these are substituted, and we are sure that the denominator is not zero. The slope of the investigated curve is wanted at Peanut. Now that we know the coordinates of the investigative point on this local curve at the given point because substitute into the immigration of the tangent climb my equals one plants one over third times X It is a line that tangent to implicitly given curve at being note.
32. Tangent lines in case of parametric functions - 1. Example: Now I'm going to show you how to determine the recreation off tangent lines. If the investigated curve is given by Parametric equations, the acts on via coordinates off the curve are given by perimeter T. You can think off a moving object on the X Y plane and T is the time the tangent line. The given function must be calculated when the time equals two pi over for the X coordinate it first to t plastic a society and the my coordinate it first of scientists. The equation of the tangent line can be written as usual. We're going to need the coordinates off the point where the tangent line touches the curve on. We're going to need the slope off the function at the given port. So far, we do not know anything, but we can easily calculate the coordinates by substituting into the aromatic creations. So let's take the equations of the coordinates and substitute pi Over 4 40 X note equals two pi over four plus co signed by over four co sand pi over four. It first to square 2/2, So the result is spyware four plus X choir to over to This is the first coordinate. The second coordinate is signed by Were for scrap Sandpiper We're for it was to square of 2/2, so rhino equals to 1/2. These are the coordinates of the investigated point. It doesn't matter how I indicate the coordinates the value off by attacks. Note it was to the value of my a t note. We just simply do not know the by ex function explicitly. That is why we use X T and variety functions. But the point is the same race. Now let's calculate the derivative off. Why, With respect to acts, we do not know the expletive connection. Which one accent? Why? So we are going to differentiate with trace back to the perimeter. The derivative off my and acts are going to be calculated with respect. The under quotient equals the derivative off private respective tax. Let's start by differentiating X'd with respect to t The derivative off T is one and the derivative of course, I NT is minus scientist. Then we can differentiate by tea with respect to t. This is a bit more complicated as scientists squad is a composite function, the counter function is taking the squared and the inner function is signed. So we get to sign if we differentiate after function and this is multiplied by derivative off Sigh Inti What together we get to 70 times CO society. Now we can calculate their a table by with respect to AXP. Why that over eggs dot first to buy Apple strove which is very But if off my meat respective packs, the result is to scientific society over one minus sanity. We have the general derivative, but we are actually looking for the slope at a given point. In case of Parametric functions, we can substitute the Param Eter value off the given point to find the slow de notice by wherefore that is substituted. Both signed fire were for and co San pie were four equals to scramble to over two. We can substitute that and calculate the value of the product in the nominator. The slope is 1/1 minus choir two or 2 30 Note. This has been the last piece of the puzzle. We can finally determine the equation of the tangent line. The general form and the required constants are known. The tangent line to The aromatic curve is 1/2 plus 1/1 minus quieter to over two times, X minus pi over four plus six. Credit to over to
33. Tangent lines in case of implicit functions - 2. Example: Let's see another example of determining tangent lines to Parametric Lee given curves. The X and vie coordinates are given as functions off t X equals. Do eat the power of to t by equals t squared plus five t The tension plan shall be determined at the point. This cramped by the permit er value T note equals 20 The creation of the tension line can be written as usual. First, we can calculate the coordinates off. The given point, he notes, can be substituted into the X T and variety functions. The X coordinate equals two e to the power of zero, which is the y Coordinate equals 20 as Tina Scranton t not are both zero. So we need the tension. Klein at the 00.10 To get the equation of the tangent line, we need the slope of the curve. At this point, in case of aromatic functions, we can calculate the derivative off why, with respective acts, by the help of the Parametric derivatives by that, and acts that other derivatives with respect to t we need down. Let's differentiate with respect to d. The axe T function is a composite function the derivative off. The exponential part is into the power of to t, and the derivative off to tea is too. Their product is the total derivative. The decorative off Rieti is to t plus five according to the derivative rule of polynomial. Therefore, the derivative off by with respect to X is still t plus 5/2 times into the power of to 30. This stands for vanity together slope. At the given point, we must substitute t note to desecration denote the 1st 20 So via derivative at axe, note it first. To do time zero plus 5/2 times e to the power of to time zero into the power of zero equals to one. So the derivative of the function is fine over to. At this given point, these equals to this local detention line. We have every necessary constant why it first to 5/2 times X minus one is the equation of the tangent line to the given priority curve. AT T note
34. Tangent lines in case of polar coordinates - 1. Example: let's practice a special case of parametric functions. This time a aromatic function is given by polar coordinates. Therefore, R is the distance from the origin off the coordinate system and fight is the angle measured from the X axis. Our is given us a function off. Five to be more precise, R equals to sign. Five. The tangent line to discover must be determined at the point described by the permit or value. Finance equals two pi over tree. The usual equation is needed on we're going to work, as we usually were with aromatic functions. However, we haven't got any angsty or writing functions yet. Instead of them, we should use the connection between the Cartesian and polar coordinates. The connection between X and Y and R and Fi are always the same. X equals two R times cause I'm fine and by equals two R times sci fi as R equals to sign Phi X equals to sign five times school side, fine wine. Why equals to sign five scrap You can always use thes e creations on the ar fi function can be substituted. By doing these steps, we get the Parametric EQ Rations X and y are functions off. Five. So fine is the perimeter. Now we can apply the same method as in case of Parametric functions began. Substitute into the creations to find the coordinates of the point described by five. Note and then we can calculate the slope of the function at the given point. Let's start by the simple substitution. Finally, it was to buy what tree so axe. Note it first to sign pi over three times cause and by over tree San pi over tree or sent 60 degrees equals two square three over to cosign pie were tree or cause an 60 degrees if us to one over to there. Productive square. 3/4. Similarly, my note equals to sign fireworks three Squad, which equals 2 3/4 A sign 60 degrees equals two square t over to so the investigated point is at square it 3/4 and 3/4. In the next step, we must differentiate the paramedic function. We need the derivative off by with respect to acts, we can get that by differentiating X environment, respect fi on taking the quotient off, violent and ex that first we can differentiate ex fight with this back to five. The product, through a monthly, applied the derivative off sci fi isco sci fi. This is multiplied by co sci fi. Then we had signed five times the derivative off go sci fi the derivative. Of course, I fight is minus sci fi. That is why you see a minus sign instead of a plus side. In case of differentiating Why fi, we deal with the composite function, the derivative off the squad terms gives us to sign Phi. This is multiplied by co sign five, which is the derivative off inner function. The quotient off the results gives us the derivative off by with respect to X denominator is via that on the denominator is accident actually the nominator it for us to sign to fire and the nominator requests to co sign to five. Therefore, the slope exhausted tangent to fi You do not need to notice this. You can also substitute fin out into this original for the slope of the curve at the given point requested the value of the derivative at feet. Now the signed off by Albert Tree is square Hootie over to and the Kusenov priority is one where two we can ride these instead of taking automatic functions we can do or the operations and get a simple result. The new military cross this choir t over to on the denominator. Three first, the minus 1/2 were together. The slope is minus Scrabble. We can calculate the results finally, as we have the X by coordinate of the investigative plant on the slope of the curve. At that given point, the attrition of the tangent client rates as my equals Dutilleux were four minus 13 times x minus square to you before.
35. Tangent lines in case of polar coordinates - 2. Example: Let's see another example with polar coordinates. The equation off the curve is R equals to fi. Where are on fire at the polar coordinates. This is a famous curve. It is called our comedian Spiral. We should determine its tangent line at final equals 25 by over two, the usually creation shall be determined. So we need the coordinates on the slope belonging. To find out the connection between the polar and Cartesian coordinates must be used first to get the Parametric EQ rations. X equals two arc time schools on Fine. Why, it was our times. Sci fi as R equals defy ax equals to fight Amsco sine Phi and Phi equals 25 times sci fi. Based on the parametric equations, we can determine the coordinates off the point belonging to find out the X coordinate zero A schools and five by over 20 you can use the period a city off consigned to get this result. The co sign off five pi over two and by over two are the same as their differences to buy and raise. The X coordinate is zero. The y coordinate is five by over two as sign five by two equals to bomb. That's calculate the slope. We have the usual formula. Based on that, we just need to determine the Parametric derivatives to find divide over the axe. We should differentiate according to the productive in case of X that the derivative off five is one. This is multiplied by co signed a fire, then the product off five and the derivative off course on fire is added. The derivative, of course, and find is minus sci fi. Similarly, by that equals to sign five plus five times CO sci fi. The derivative off five is one and there but it off sign is co sign the derivative off by with respect to acts is the quotient off? Why that and ex that this is actually rallied. If the denominator is not zero, if it would be zero at a point with have a little terrible. Fortunately, the denominator is not zero. In our case, five by over two can be substituted for five the value off the signs are from, and the values of the co sign our zero. So there is a bomb plot zero in the nominator and zero minus five pi over two in the denominator in a simpler form. The result is minus do over five. By that is the slope of the curve. At a given point, all of the necessary constants are none. The equation of the tangent line can be determined by substitution my equals 25 by two miners to over 55 times acts.
36. Taylor polynomial: let me show you a generalization Off the tangent lines. The Taylor polynomial is a great tool off function approximation. You can see the Taylor polynomial in two different but every violent forms. T is the Taylor Polina meal off the F function around X note. So this bull in a meal is an approximation off the F function around the given axe knelt point in the small area off this point, the approximation can be very good, even though there are simple terms in the approximation, the kids derivative off F appears in the some, along with que factorial and the distance from axe knelt at the property power. If you would like to get an ant order approximation, you must differentiate half an times. Then you can substitute X knelt into the derivatives and calculate the terms off. The some in the first formula included F. Ax knelt into the some as a zero derivative. It is better to see that apart outside off the sun, so the some goes from one to N, and today's on the terms, including Terry Bodies. I mentioned that the Taylor polynomial is a generalization off the tangent line for an in close to one. The first order Taylor polynomial, is the tangent line. The tangent line is already an approximation of the curve. This is generalized by using higher order polynomial us the ant order data polynomial touches off at aunt order The tension Klein only touches in first order. Actually, this sentence means that the finger Polina contents terms tangent to the curve the derivative of the curve and so on until the anti derivative of the curve. Why is it good? Is it good? It is a great thing as derivatives describe the shape of the function as the tailor pull in . Um, EL also takes the derivatives in account. The approximation becomes really good around. Excellent. Just as a remark. Generally, two functions touches each other at ants order if they equal accident and also their very bodies are equal opportunity and order. If an plus one derivatives are a further functions are touching each other at an plus wants order. The ants or their state are swollen. Um, Ian touches f at ants order and it can be checked based on its definition in the race. For you, the approximation can be important and know the exact theoretical background. When I say that the approximation is good around X note, I should really at some reference to death. F X can be written as the sum of its data for in a meal and remaining part that is different. Similarly, as the UN plus ones or their term off the approximation. This is one way of describing the error off approximation. The error is the difference of FX function and the tea and approximation. This error can be estimated as it is always smoother than the results on the right hand side. If C is between axe, note and acts here, X is the current value off acts that is substituted into half on axe. Knelt is the point around which the approximation is down. It's not easy to get an exact upper limit of the error, but you can see the magnitude off the error for an ant order. Taylor polynomial. The error is one order higher. If ax is close to axe note. The error is very small as the father terms are even smaller than X minus accident. However, if X minus accident is greater than bomb, so X is quite far from axe note. The approximation can get really bad, so this is just a local approximation. It has a history of significance, and you could even use them for solving differential equations. But in engineering practice, you can usually meet with approximations based on fuel ear Siri's. The four year series uses trigger automatic polynomial is the full meals, but that topic is for another day.
37. Taylor polynomial - 1. Example: let me show you how to determine a Taylor polynomial. The third order trailer pulling a meal off E to the power of two X sharp determined around XML equals 20 So we are going to approximate the exponential function by 1/3 order polynomial in the general definition. On a first victory, there are going to be three terms from the some to get those we needed to differentiate acts three times. Then we can substitute accident into every derivative faction. But first we can just absolute accident into affects. The result is one as he to the power of zero is far. Now we can start differentiating we have a composite function. The other part is exponential. The inner is to axe the derivative of actual ensure part gives us e to the power of to acts on the derivative of two acts is too. So the first derivative is two times either. The patter of two acts began Substitute zero for X two times into the power of zero. Exhausted too. The second derivative equals to four times e to the power of to acts. To get this, we differentiate the first derivative by X the derivative off each of the power of to acts is two times e to the power of two X. This is multiplied by the constant multiplier To buy substitution we get for at X not we need one more derivative. The third derivative becomes eight times e to the power of two x Similarly to the previous derivatives, We should just multiplied by two by substitution. The third derivative is eight at accident. Finally, we knew everything that we need to substitute into the general formula f vaccine lt's one than the first derivative that accident is too. This is divided by a bomb factory on Andi multiplied by the first power of X minus accident Here accident equals 20 The second derivative is for that accident. This is divided by two factorial on multiplied by the second power of X minus accident. Finally, the third derivative is eight at accident it is divided by three factorial which equals 26 on. Then it is multiplied by the third power off X minus accident. We can do some re formulations if we want, but we already have the result. Tea tree is the third order putting a meal that approximately eat 1st 2 f x Iran zero
38. Taylor polynomial - 2. Example: Let's see another example. The second order Taylor polynomial off San X plus X minus pi squared shall be determined around accident requested by So he crossed it to in the general form and we only need to differentiate device. Let's start by substituting accident into F ax signed by equals zero and also by minus pi equals 20 So the function it 1st 0 at X. Now let's differentiate the derivative off. Sine X is co sign X The derivative off X minus pi squared is two times x minus pi times bomb. Actually, it is a composite function, but you can't really notice it in the result as the deputy off x minus by the inner function is by substitution, we get minus one. As a result, co signed by is minus one x minus pi is steer zero if acts first in the next step, the derivative off cosign X is minus sine x and the derivative off to acts is to and the derivative off minus two. Pi is zero. The second derivative equals to two at axe Note. As sci fi equals 20 Now we can substitute into the general formula. F X not zero. The first derivative is minus one at accident. This is divided by home factorial on multiplied by X minus axe Note. Our accident equals two pi. The second derivative is to at X note. This is divided by two factorial on multiplied by X minus pi scrap. This is very much straightforward. Let me just one more terror tickle thing. We can determine the result term by term, meaning that the Dana pulling them off. F X requested a some off the table point Amiel off sine X and the Taylor polynomial off X minus spice crab. I don't want to bore you with differentiating again. You can simply differentiate sine x by itself and you can substitute into the derivatives. The second order Teodor polynomial of signed acts around by is minus 1/1 times X minus by the bigger Polina meal off X minus by scratch is simple. It is itself X minus. Spice crab is potato Polina meal as only terms like X minus x note to an integer power appear. In such cases, you can say that the polynomial is a taylor polynomial By the arranging Polina meals, you could get Taylor Polina meals like form in this way, you can just avoid def initiation. Now, the second order pulling the meal is simply a function off itself. The function was X minus pi to the third barber. It couldn't be a second order approximation. The second order approximation off the third order X minus park you'd would be zero. You can check this by using the definition the Taylor polynomial of F acts does equal to the sum off the table for Nahmias. Just remember that you can simplify the method if you need to work with meals.
39. Thank you!: Congratulations. You have reached the end of the scores. Thank you for your attention and for staying with me near the end of the course. I'm happy that you choose to learn with me. And I hope you have enjoyed the lectures. If you haven't completed the class project yet, go ahead and try to solve the problems on your own. If you find the challenging task you can share with the others, we can solve it together. See you around.