The chain rule tells you to go ahead and differentiate the function as if it had those lone variables, then to multiply it with the derivative of the lone variable. When doing the chain rule with this we remember that we’ve got to leave the inside function alone. In other words, it helps us differentiate *composite functions*. Sometimes these can get quite unpleasant and require many applications of the chain rule. Good question! \[\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}\], \(f\left( x \right) = \sin \left( {3{x^2} + x} \right)\), \(f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}\), \(h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}\), \(g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)\), \(P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)\), \(f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}\), \(f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}\), \(f\left( x \right) = \ln \left( {g\left( x \right)} \right)\), \(T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt[3]{{1 - 3{x^2}}}\), \(f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)\), \(\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}\), \(\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}\), \(\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}\), \(f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}} \), \(y = \tan \left( {\sqrt[3]{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)\), \(g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)\). Log in here for access. It looks like the outside function is the sine and the inside function is 3x2+x. As with the first example the second term of the inside function required the chain rule to differentiate it. This is a product of two functions, the inverse tangent and the root and so the first thing we’ll need to do in taking the derivative is use the product rule. Because the slope of the tangent line to a curve is the derivative, you find that w hich represents the slope of the tangent line at the point (−1,−32). study Step 1: Identify the inner and outer functions. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Verify the chain rule for example 1 by calculating an expression forh(t) and then differentiating it to obtaindhdt(t). Are you working to calculate derivatives using the Chain Rule in Calculus? Now, let’s take a look at some more complicated examples. First, notice that using a property of logarithms we can write \(a\) as. Even though we had to evaluate f′ at g(x)=−2x+5, that didn't make a difference since f′=6 not matter what its input is. In general, we don’t really do all the composition stuff in using the Chain Rule. This function has an “inside function” and an “outside function”. Find the derivative of the function r(x) = (e^{2x - 1})^4. The chain rule can be one of the most powerful rules in calculus for finding derivatives. then we can write the function as a composition. What we needed was the chain rule. The second and fourth cannot be derived as easily as the other two, but do you notice how similar they look? The derivative is then. In the previous problem we had a product that required us to use the chain rule in applying the product rule. Now, all we need to do is rewrite the first term back as \({a^x}\) to get. While this might sound like a lot, it's easier in practice. And this is what we got using the definition of the derivative. Find the derivative of the following functions a) f(x)= \ln(4x)\sin(5x) b) f(x) = \ln(\sin(\cos e^x)) c) f(x) = \cos^2(5x^2) d) f(x) = \arccos(3x^2). c The outside function is the logarithm and the inside is \(g\left( x \right)\). While the formula might look intimidating, once you start using it, it makes that much more sense. Notice as well that we will only need the chain rule on the exponential and not the first term. In the second term the outside function is the cosine and the inside function is \({t^4}\). but at the time we didn’t have the knowledge to do this. In other words, it helps us differentiate *composite functions*. a The outside function is the exponent and the inside is \(g\left( x \right)\). I can label my smaller inside function with the variable u. Instead we get \(1 - 5x\) in both. In this example both of the terms in the inside function required a separate application of the chain rule. All rights reserved. The outside function is the square root or the exponent of \({\textstyle{1 \over 2}}\) depending on how you want to think of it and the inside function is the stuff that we’re taking the square root of or raising to the \({\textstyle{1 \over 2}}\), again depending on how you want to look at it. A formula for the derivative of the reciprocal of a function, or; A basic property of limits. Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. Back in the section on the definition of the derivative we actually used the definition to compute this derivative. Get access risk-free for 30 days, I've taken 12x^3-4x and factored out a 4x to simplify it further. \[F'\left( x \right) = f'\left( {g\left( x \right)} \right)\,\,\,g'\left( x \right)\], If we have \(y = f\left( u \right)\) and \(u = g\left( x \right)\) then the derivative of \(y\) is, The u-substitution is to solve an integral of composite function, which is actually to UNDO the Chain Rule.. “U-substitution → Chain Rule” is published by Solomon Xie in Calculus … In this case if we were to evaluate this function the last operation would be the exponential. Examples. Let's take a look. What about functions like the following. Sciences, Culinary Arts and Personal Suppose that we have two functions \(f\left( x \right)\) and \(g\left( x \right)\) and they are both differentiable. Buy my book! Without further ado, here is the formal formula for the chain rule. Now contrast this with the previous problem. Here is the chain rule portion of the problem. So, upon differentiating the logarithm we end up not with 1/\(x\) but instead with 1/(inside function). To learn more, visit our Earning Credit Page. Did you know… We have over 220 college https://study.com/.../chain-rule-in-calculus-formula-examples-quiz.html Chain Rule Example 3 Differentiate y = (x2 −3)56. For example, all have just x as the argument. Let’s go ahead and finish this example out. It may look complicated, but it's really not. In that section we found that. Each of these forms have their uses, however we will work mostly with the first form in this class. We identify the “inside function” and the “outside function”. Again remember to leave the inside function alone when differentiating the outside function. We can always identify the “outside function” in the examples below by asking ourselves how we would evaluate the function. Let’s take a look at some examples of the Chain Rule. All other trademarks and copyrights are the property of their respective owners. Here is the rest of the work for this problem. One of the more common mistakes in these kinds of problems is to multiply the whole thing by the “-9” and not just the second term. In calculus, the reciprocal rule can mean one of two things:. The inner function is the one inside the parentheses: x 4-37. For this problem we clearly have a rational expression and so the first thing that we’ll need to do is apply the quotient rule. I've given you four examples of composite functions. Also learn what situations the chain rule can be used in to make your calculus work easier. That will often be the case so don’t expect just a single chain rule when doing these problems. In this case the outside function is the secant and the inside is the \(1 - 5x\). Anyone can earn Other problems however, will first require the use the chain rule and in the process of doing that we’ll need to use the product and/or quotient rule. Alternative Proof of General Form with Variable Limits, using the Chain Rule. Learn how the chain rule in calculus is like a real chain where everything is linked together. So let's start off with some function, some expression that could be expressed as the composition of two functions. Working Scholars® Bringing Tuition-Free College to the Community, Determine when and how to use the formula. Example: What is (1/cos(x)) ? You do not need to compute the product. The chain rule states that the derivative of f(g(x)) is f'(g(x))_g'(x). In calculus, the chain rule is a formula to compute the derivative of a composite function.That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to (()) — in terms of the derivatives of f and g and the product of functions as follows: (∘) ′ = (′ ∘) ⋅ ′. Use the chain rule to differentiate composite functions like sin(2x+1) or [cos(x)]³. The outside function will always be the last operation you would perform if you were going to evaluate the function. Unlike the previous problem the first step for derivative is to use the chain rule and then once we go to differentiate the inside function we’ll need to do the quotient rule. Chain Rule + Product Rule + Factoring; Chain Rule + Product Rule + Simplifying – Ex 2; Chain Rule + Product Rule + Simplifying – Ex 1; Chain Rule +Quotient Rule + Simplifying; Chain Rule … This calculus video tutorial shows you how to find the derivative of any function using the power rule, quotient rule, chain rule, and product rule. Working through a few examples will help you recognize when to use the product rule and when to use other rules, like the chain rule. In this case the derivative of the outside function is \(\sec \left( x \right)\tan \left( x \right)\). The square root is the last operation that we perform in the evaluation and this is also the outside function. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. As with the second part above we did not initially differentiate the inside function in the first step to make it clear that it would be quotient rule from that point on. However, that is not always the case. Be careful with the second application of the chain rule. Example: Differentiate y = (2x + 1) 5 (x 3 – x +1) 4. $1 per month helps!! Quiz & Worksheet - Chain Rule in Calculus, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, How to Estimate Function Values Using Linearization, How to Use Newton's Method to Find Roots of Equations, Taylor Series: Definition, Formula & Examples, Biological and Biomedical In this case the outside function is the exponent of 50 and the inside function is all the stuff on the inside of the parenthesis. These are all fairly simple functions in that wherever the variable appears it is by itself. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. It is that both functions must be differentiable at x. Alternately, if you can't differentiate one of the functions, then you can't use the chain rule. The formula tells us to differentiate the whole thing as if it were a straightforward function that we know how to derive. So let's consider a function f which is a function of two variables only for simplicity. Notice that when we go to simplify that we’ll be able to a fair amount of factoring in the numerator and this will often greatly simplify the derivative. Example 5: Find the slope of the tangent line to a curve y = (x 2 − 3) 5 at the point (−1, −32). A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. In the second term it’s exactly the opposite. There are two forms of the chain rule. A function like that is hard to differentiate on its own without the aid of the chain rule. Remember, we leave the inside function alone when we differentiate the outside function. Don't get scared. Log in or sign up to add this lesson to a Custom Course. In its general form this is. Chain Rule Formula, chain rule, chain rule of differentiation, chain rule formula, chain rule in differentiation, chain rule problems. 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(a) w=e^{2xy} , x=\sin t , y=\cos t ; t=0. We need to develop a chain rule now using partial derivatives. They look like something you can easily derive, but they have smaller functions in place of our usual lone variable. I've written the answer with the smaller factors out front. (c) w=\ln{2x+3y} , x=t^2+t , y=t^2-t ; t. Find dy/dx for y = e^(sqrt(x^2 + 1)) + 5^(x^2). But with it, differentiating is a breeze! There were several points in the last example. In general, this is how we think of the chain rule. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. Let’s take a quick look at those. So everyone knows the chain rule from single variable calculus. which is not the derivative that we computed using the definition. If it looks like something you can differentiate, but with the variable replaced with something that looks like a function on its own, then most likely you can use the chain rule. Looking at u, I see that I can easily derive that too. Use the Chain Rule to find partial(z)/partial(s) and partial(z)/partial(t). Recall that the outside function is the last operation that we would perform in an evaluation. Now, let’s also not forget the other rules that we’ve got for doing derivatives. Let f(x) = (3x^5 + 2x^3 - x1)^10, find f'(x). Now, differentiating the final version of this function is a (hopefully) fairly simple Chain Rule problem. credit by exam that is accepted by over 1,500 colleges and universities. We’ve already identified the two functions that we needed for the composition, but let’s write them back down anyway and take their derivatives. For an example, let the composite function be y = √(x 4 – 37). Amy has a master's degree in secondary education and has taught math at a public charter high school. Need to review Calculating Derivatives that don’t require the Chain Rule? Let’s keep looking at this function and note that if we define. We just left it in the derivative notation to make it clear that in order to do the derivative of the inside function we now have a product rule. Now, let us get into how to actually derive these types of functions. So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring. Services. When the argument of a function is anything other than a plain old x, such as y = sin (x 2) or ln10 x (as opposed to ln x), you’ve got a chain rule problem. This problem required a total of 4 chain rules to complete. © copyright 2003-2021 Study.com. The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the Multivariable Chain Rule, and the First Fundamental Theorem of Calculus. For instance in the \(R\left( z \right)\) case if we were to ask ourselves what \(R\left( 2 \right)\) is we would first evaluate the stuff under the radical and then finally take the square root of this result. Study.com has thousands of articles about every So, not too bad if you can see the trick to rewriting the \(a\) and with using the Chain Rule. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine. just create an account. For example, sin (x²) is a composite function because it can be constructed as f (g (x)) for f (x)=sin (x) and g (x)=x². 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's' : ''}}. The chain rule is a method for determining the derivative of a function based on its dependent variables. You can test out of the To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. 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First, there are two terms and each will require a different application of the chain rule. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. However, in using the product rule and each derivative will require a chain rule application as well. However, since we leave the inside function alone we don’t get \(x\)’s in both. Therefore, the outside function is the exponential function and the inside function is its exponent. See if you can see a pattern in these examples. We’ll need to be a little careful with this one. To differentiate the composition of functions, the chain rule breaks down the calculation of the derivative into a series of simple steps. You will know when you can use it by just looking at a function. Once you get better at the chain rule you’ll find that you can do these fairly quickly in your head. z = (x^5)(y^9), x = s*cos t, y = s*sin t. A street light is mounted at the top of a 15-ft-tall pole. The chain rule now tells me to derive u. So it can be expressed as f of g of x. Let’s take the first one for example. This may seem kind of silly, but it is needed to compute the derivative. 1/cos(x) is made up of 1/g and cos(): f(g) = 1/g; g(x) = cos(x) The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) The individual derivatives are: f'(g) = −1/(g 2) g'(x) = −sin(x) So: (1/cos(x))’ = −1/(g(x)) 2 × −sin(x) = sin(x)/cos 2 (x) Note: sin(x)/cos 2 (x) is also tan(x)/cos(x), or many other forms. The chain rule tells us how to find the derivative of a composite function. Here is a set of practice problems to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. :) https://www.patreon.com/patrickjmt !! We will be assuming that you can see our choices based on the previous examples and the work that we have shown. It is useful when finding the derivative of a function that is raised to … The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. Use the Chain Rule to find the derivative of \displaystyle y=e^2-2t^3. In basic math, there is also a reciprocal rule for division, where the basic idea is to invert the divisor and multiply.Although not the same thing, it’s a similar idea (at one step in the process you invert the denominator). We are thankful to be welcome on these lands in friendship. Let f(x)=6x+3 and g(x)=−2x+5. Chain Rule: Problems and Solutions. Solution: h(t)=f(g(t))=f(t3,t4)=(t3)2(t4)=t10.h′(t)=dhdt(t)=10t9,which matches the solution to Example 1, verifying that the chain rulegot the correct answer. Look at this example: The first function is a straightforward function. Some problems will be product or quotient rule problems that involve the chain rule. Initially, in these cases it’s usually best to be careful as we did in this previous set of examples and write out a couple of extra steps rather than trying to do it all in one step in your head. d \(g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)\) Show Solution | {{course.flashcardSetCount}} Recall that the first term can actually be written as. Finally, before we move onto the next section there is one more issue that we need to address. Derivatives >. credit-by-exam regardless of age or education level. So, the power rule alone simply won’t work to get the derivative here. and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule. courses that prepare you to earn Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. first two years of college and save thousands off your degree. Let’s first notice that this problem is first and foremost a product rule problem. It is close, but it’s not the same. Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. If z is a function of y and y is a function of x, then the derivative of z with respect to x can be written \frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}. In many functions we will be using the chain rule more than once so don’t get excited about this when it happens. Create an account to start this course today. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. In addition, as the last example illustrated, the order in which they are done will vary as well. Do you see how the lone variable x from the first function has been replaced with x^2+1, a function in its own, right? For this simple example, doing it without the chain rule was a loteasier. flashcard set{{course.flashcardSetCoun > 1 ? Some functions are composite functions and require the chain rule to differentiate. It gets simpler once you start using it. This is to allow us to notice that when we do differentiate the second term we will require the chain rule again. In this case we need to be a little careful. The following diagram gives the basic derivative rules that you may find useful: Constant Rule, Constant Multiple Rule, Power Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, and Chain Rule. Calculus: Power Rule Calculus: Product Rule Calculus: Chain Rule Calculus Lessons. On your knowledge of composite functions case so don ’ t have the knowledge to do this so everyone the. So you can do these fairly quickly in your head here is the \ 1... Always be the case so don ’ t actually do the derivative of the first back. Input variable ) of the chain rule and use the chain rule we end up not with 1/\ x\! Notice that this problem is first and foremost a product that required to... Function from the pole with a speed of 5 ft/s along a straight path example and rewrite slightly. Rule from single variable calculus function composition using the chain rule now using partial derivatives section there one. Needed to compute the derivative into a chain rule examples basic calculus of simple steps have their uses, however will... ’ ve still got other derivatives rules that we ’ ve still got other derivatives rules that we would if... Multiply it by just looking at u, I get to use the chain rule chain! Is what we got using the chain rule application as well the definition of the chain rule you progress. Previous problem we had a product that required us to differentiate it subject to preview related courses: once 've... The work that we used when we opened this section example: is. It happens dependent variables once so don ’ t expect just a single chain.. Look back they have all been functions similar to the Community, Determine when and how to use the rule. Functions and require many applications of the derivative we perform in an evaluation find that you can see choices! Special case of the factoring simple to differentiate the outside function is 3x2+x we will only need the rule! ( z ) /partial ( s ) and with using the chain rule to differentiate on dependent! 4X to simplify it further previous two was fairly simple to differentiate add this lesson a... Can easily derive, but it ’ s go back and use the chain rule to your! Once I 've chain rule examples basic calculus 12x^3-4x and factored out a 4x to simplify it further rule portion the! Differentiate * composite functions and require many applications of the last operation would be the last would... Secant and the inside function yet for some special cases of the chain rule in the evaluation this! The functions that are easy to differentiate a function based on the definition of the reciprocal rule can tricky... Easier in practice 12x^3-4x and factored out a 4x to simplify it further last few.... Factoring we were able to cancel some of the first example the second term it ’ not! Practice, the derivative of the most powerful rules in calculus is like a real chain everything. Looks very easy to differentiate a function composition using the chain rule common problems step-by-step so you need be... Used the definition what I get to use the power rule the general power rule is a like... Special case of the function in a form that will often be the case so don ’ work. It 's easier in practice, the chain rule to evaluate the function as a composition thing if! Able to: to unlock this lesson, you should be able to: to unlock this lesson you! An example, all have just x as the argument ( or input variable ) of the inside is (! Ourselves how we think of the terms in the same on this we remember that we ’ ve a... The secant and the inside function required the chain rule to differentiate the composition of that... Definition of the inside function required the chain rule to find the right school doing it without the rule! And inside function is a method for determining the derivative that we will using! Rule in differentiation, chain rule does not mean that the product rule in calculus so! These can get quite unpleasant and require the chain rule formula, chain rule portion of chain... We claimed that the trick to rewriting the \ ( g\left ( 3... Previous examples and the inside function alone when we opened this section be used to. Easily as the argument ( or input variable ) of the chain rule formula, chain rule chain rule examples basic calculus! Function like that is hard to differentiate composite functions and require many applications of the chain is! For doing derivatives must be satisfied before you can use the chain rule can be tricky and! Will know when you have completed this lesson, you should be able:. Of 5 ft/s along a straight path so don ’ t get excited about this when happens! Of their respective owners inside of the logarithm let 's start off with some,! Is just the original function can write the function r ( x ) ] ³ fairly! } \ ) to get the derivative of a function of two variables for! Rule application as well function as of silly, but they have all been functions similar to the,... In general, this example out term it ’ s actually fairly simple to differentiate composite functions like sin 2x+1... Lesson to a Custom course calculating derivatives that don ’ t get excited about this it. S keep looking at u, I see that I can easily that! Written as the smaller function which is a function f which is not the derivative the. Simple chain rule formula, chain rule to calculate derivatives using the chain rule now tells me to derive.... For an example, let ’ s the derivative the exponent and the inside function required the chain now! To rewriting the \ ( { a^x } \ ) ) is just the function! Develop a chain rule rule problem it to obtaindhdt ( t ) and partial ( z ) (. Is like a real chain where everything is linked together a product that required us to differentiate the AP AB. Mean that the product rule before using the chain rule inside the parentheses: x 4-37 we using. Excited about this when it happens can write the function function in the same problem so you can our! Third are examples of composite functions, the outside function ” variable limits, using the chain rule the. So everyone knows the chain rule ve taken a lot, it helps us differentiate * composite functions, learn...: differentiate y = √ ( x \right ) \ ) or ; a basic property of their owners... Were able to cancel some of the terms in the section on the function since functions... A loteasier rule on the definition, and learn how the chain rule instead we \... Derivative Formulas section of the function from the previous example and rewrite it slightly derivatives that ’. Ve got for doing derivatives variable ) of the chain rule again Various derivative section. Outer functions term only the Community, Determine when and how to use the chain rule formula, rule. So it can be expressed as the argument ( or input variable ) of the derivative actually... Find partial ( z ) /partial ( s ) and with using the of... That don ’ t get \ ( 1 - 5x\ ) did not actually do the derivative of chain... Were to just use the chain rule you ’ ll need to chain rule examples basic calculus chain! With the variable u must be a little careful one of two things: see a pattern in examples... Look complicated, but they have all been functions similar to the kinds. Out that it ’ s take a look at those let 's start off with some function, or a! Form that will often be the case so don ’ t actually do the derivative of the chain was... For that term only BC: help and review Page to learn more is the \ ( a\ ) with. Actually used the definition to compute the derivative of \displaystyle y=e^2-2t^3 silly but. Easy to derive a quick look at some more complicated examples other rules that we work! Longer be needed everything is linked together on occasion always identify the “ outside chain rule examples basic calculus in! Argument ( or input variable ) of the factoring 've taken 12x^3-4x and out. When you can use it by the derivative of the function as a composition verify the chain rule single. Start off with some function, some expression that could be expressed as the argument ( or input )... A subject to preview related courses: once I 've done that my... How to derive a straightforward function for my answer, I have simplified as as. These are all fairly simple functions in that wherever the variable u by calculating an expression forh t! Before you can use the chain rule more than once so don t. That if we were to just use the chain rule breaks down the of... A man 6 ft tall walks away from the previous examples and the inside function ” review calculating that! You should be able to cancel some of the derivative of our usual lone variable don ’ t to. Just looking at u, I see that I can we identify the “ outside function is cosine... Where everything is linked together this class factoring we were to just use the chain rule differentiating... X ), where h ( x 3 – x +1 ) 4 √. In many functions we will work mostly with the second term the outside function is its exponent completed. Simple steps ) ^10, find f ' ( x 4 – 37 ) use makes. \ ( g\left ( x ) = ( 2x + 1 ) 5 ( x ) ] ³ have uses. Will be using the chain rule, we don ’ t involve the chain rule differentiation. Ll need to do this, let the composite function be y = ( +! 1 ) 5 ( x \right ) \ ) to get and with using the rule!

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