There were several points in the last example. but at the time we didn’t have the knowledge to do this. In short, we would use the Chain Rule when we are asked to find the derivative of function that is a composition of two functions, or in other terms, when we are dealing with a function within a function. That can get a little complicated and in fact obscures the fact that there is a quick and easy way of remembering the chain rule that doesn’t require us to think in terms of function composition. Recall that the first term can actually be written as. Here they are. We then differentiate the outside function leaving the inside function alone and multiply all of this by the derivative of the inside function. If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. which is not the derivative that we computed using the definition. In this part be careful with the inverse tangent. However, if you look back they have all been functions similar to the following kinds of functions. Get Better In this problem we will first need to apply the chain rule and when we go to differentiate the inside function we’ll need to use the product rule. Whenever the argument of a function is anything other than a plain old x, you’ve got a composite […] In addition, as the last example illustrated, the order in which they are done will vary as well. For instance, if you had sin(x^2 + 3) instead of sin(x), that would require the chain rule. We use the chain rule when differentiating a 'function of a function', like f (g (x)) in general. First, notice that using a property of logarithms we can write $$a$$ as. It is useful when finding the derivative of e raised to the power of a 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 And I'll have a special version of the chain rule that I'll use for these and I'll call this rule the general exponential rule. 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. This is to allow us to notice that when we do differentiate the second term we will require the chain rule again. Let us find the derivative of . and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule. The composition of two functions $f$ with $g$ is denoted $f\circ g$ and it's defined by $(f\circ g)(x)=f(g(x)). In the process of using the quotient rule we’ll need to use the chain rule when differentiating the numerator and denominator. Use the product rule when you have a product. Other problems however, will first require the use the chain rule and in the process of doing that we’ll need to use the product and/or quotient rule. The chain rule is a formula to calculate the derivative of a composition of functions. If you're seeing this message, it means we're having trouble loading external resources on our website. For example, you would use it to differentiate sin(3x) (With the function 3x being inside the sin() function) In its general form this is. Let’s first notice that this problem is first and foremost a product rule problem. Notice that when we go to simplify that we’ll be able to a fair amount of factoring in the numerator and this will often greatly simplify the derivative. Example problem: Differentiate y = 2 cot x using the chain rule. This may seem kind of silly, but it is needed to compute the derivative. When the chain rule comes to mind, we often think of the chain rule we use when deriving a function. Second, we need to be very careful in choosing the outside and inside function for each term. Let’s take the function from the previous example and rewrite it slightly. The composition of two functions [math]f$ with $g$ is denoted $f\circ g$ and it's defined by [math](f\circ g Chain rule is also often used with quotient rule. The chain rule tells us how to find the derivative of a composite function. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine. These are all fairly simple functions in that wherever the variable appears it is by itself. In this case the derivative of the outside function is $$\sec \left( x \right)\tan \left( x \right)$$. For the most part we’ll not be explicitly identifying the inside and outside functions for the remainder of the problems in this section. I want to talk about a special case of the chain rule where the function that we're differentiating has its outside function e to the x so in the next few problems we're going to have functions of this type which I call general exponential functions. It looks like the outside function is the sine and the inside function is 3x2+x. Let’s take a quick look at those. Some problems will be product or quotient rule problems that involve the chain rule. The answer is given by the Chain Rule. However, in practice they will often be in the same problem so you need to be prepared for these kinds of problems. The chain rule says that So all we need to do is to multiply dy /du by … • Solution 2. 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. start your free trial. Finally, before we move onto the next section there is one more issue that we need to address. Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex Let’s go ahead and finish this example out. For example, if a composite function f( x) is defined as The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. There are two forms of the chain rule. Before we discuss the Chain Rule formula, let us give another example. So, the power rule alone simply won’t work to get the derivative here. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. Since the functions were linear, this example was trivial. But it's always ignored that even y=x^2 can be separated into a composition of 2 functions. And this is because the derivative of e to the x if you'll recall derivative of e to the x is just e to the x. We could of course simplify the result algebraically to $14x(x^2+1)^2,$ but we’re leaving the result as written to emphasize the Chain rule term $2x$ at the end. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. In many functions we will be using the chain rule more than once so don’t get excited about this when it happens. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… 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. MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). That means that where we have the $${x^2}$$ in the derivative of $${\tan ^{ - 1}}x$$ we will need to have $${\left( {{\mbox{inside function}}} \right)^2}$$. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? To put this rule into context, let’s take a look at an example: $$h(x)=\sin(x^3)$$. Let's keep it simple and just use the chain rule and quotient rule. So Deasy over D s. Well, we see that Z depends on our in data. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. If we were to just use the power rule on this we would get. What we needed was the chain rule. Step 1 Rewrite the function in terms of the cosine. That will often be the case so don’t expect just a single chain rule when doing these problems. And this is because the derivative of e to the x if you'll recall derivative of e to the x is just e to the x. Remember, we leave the inside function alone when we differentiate the outside function. Section 2-6 : Chain Rule We’ve been using the standard chain rule for functions of one variable throughout the last couple of sections. $\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}$, $$f\left( x \right) = \sin \left( {3{x^2} + x} \right)$$, $$f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}$$, $$h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}$$, $$g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)$$, $$P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)$$, $$f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}$$, $$f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}$$, $$f\left( x \right) = \ln \left( {g\left( x \right)} \right)$$, $$T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt{{1 - 3{x^2}}}$$, $$f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)$$, $$\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}$$, $$\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}$$, $$\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}$$, $$f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}}$$, $$y = \tan \left( {\sqrt{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)$$, $$g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)$$. The chain rule is also used when you want to differentiate a function inside of another function. In this case, you could debate which one is faster. In general, we don’t really do all the composition stuff in using the Chain Rule. It is close, but it’s not the same. Current time:0:00Total duration:2:27. We identify the “inside function” and the “outside function”. Application, Who The chain rule is a rule for differentiating compositions of functions. So how do you differentiate one these well we're going to use a version of the chain rule that I'm calling the general power rule. 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. The chain rule works for several variables (a depends on b depends on c), just propagate the wiggle as you go. We The chain rule isn't just factor-label unit cancellation -- it's the propagation of a wiggle, which gets adjusted at each step. 11 Partial derivatives and multivariable chain rule 11.1 Basic deﬁntions and the Increment Theorem One thing I would like to point out is that you’ve been taking partial derivatives all your calculus-life. As with the first example the second term of the inside function required the chain rule to differentiate it. Again remember to leave the inside function alone when differentiating the outside function. Therefore, the outside function is the exponential function and the inside function is its exponent. 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. He still trains and competes occasionally, despite his busy schedule. To illustrate this, if we were asked to differentiate the function: … In that section we found that. It’s also one of the most important, and it’s used all the time, so … more. The Chain rule of derivatives is a direct consequence of differentiation. So, not too bad if you can see the trick to rewriting the $$a$$ and with using the Chain Rule. Click HERE to return to the list of problems. 2 Exercise 3.4.19 Prove that d dx cotx = −csc2 x. Example. c The outside function is the logarithm and the inside is $$g\left( x \right)$$. Now, the chain rule is a little bit tricky to get a hang of at first, and this video does a great job of showing you the process. In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. The exponential rule is a special case of the chain rule. If it looks like something you can differentiate When doing the chain rule with this we remember that we’ve got to leave the inside function alone. Just skip to 4:40 in the video for a chain rule lesson. In this case if we were to evaluate this function the last operation would be the exponential. Step 1 Differentiate the outer function. In calculus, the chain rule is a formula to compute the derivative of a composite function. You should only need to use the limit definition if you have a strangely-defined function that your can't use the rules for, such as a weird piecewise function. The chain rule can be thought of as taking the derivative of the outer function (applied to the inner function) and multiplying it times the derivative of the inner function. Now contrast this with the previous problem. Be careful with the second application of the chain rule. Practice: Chain rule capstone. Implicit differentiation. The chain rule is used to find the derivative of the composition of two functions. As another example, e sin x is comprised of the inner function sin The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. sinx.cosx, where you have two distinct functions, you can use chain rule but product rule is quicker. General Power Rule a special case of the Chain Rule. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, If we define $$F\left( x \right) = \left( {f \circ g} \right)\left( x \right)$$ then the derivative of $$F\left( x \right)$$ is, Eg: 45x^2/ (3x+4) Similarly, there are two functions here plus, there is a denominator so you must use the Quotient rule to differentiate. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. The Chain and Power Rules Combined We can now apply the chain rule to composite functions, but note that we often need to use it with other rules. There are a couple of general formulas that we can get for some special cases of the chain rule. Once you have a grasp of the basic idea behind the chain rule, the next step is to try your hand at some examples. 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. Proving the chain rule. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x). Recall that the chain rule states that . Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. Now, using this we can write the function as. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. So, upon differentiating the logarithm we end up not with 1/$$x$$ but instead with 1/(inside function). (4 votes) Steps for using chain rule, and chain rule with substitution. We’ve taken a lot of derivatives over the course of the last few sections. a composite function). 1. 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. 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. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. Notice that we didn’t actually do the derivative of the inside function yet. In this case we did not actually do the derivative of the inside yet. Let’s take the first one for example. Let’s use the second form of the Chain rule above: Are, Learn By the way, here’s one way to quickly recognize a composite function. Indeed, we have So we will use the product formula to get The chain rule is often one of the hardest concepts for calculus students to understand. We use the product rule when differentiating two functions multiplied together, like f (x)g (x) in general. So, the derivative of the exponential function (with the inside left alone) is just the original function. The chain rule is for differentiating a function that is composed of other functions in a particular way (i.e. In the second term it’s exactly the opposite. #y=((1+x)/(1-x))^3=((1+x)(1-x)^-1)^3=(1+x)^3(1-x)^-3# 3) You could multiply out everything, which takes a bunch of time, and then just use the quotient rule. Use the chain rule to find $$\displaystyle \frac d {dx}\left(\sec x\right)$$. we'll have e to the x as our outside function and some other function g of x as the inside function.And I'll have a special version of the chain rule that I'll use for these and I'll call this rule the general exponential rule. Sometimes these can get quite unpleasant and require many applications of the chain rule. Example. 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. To unlock all 5,300 videos, The outside function will always be the last operation you would perform if you were going to evaluate the function. Here is the rest of the work for this problem. Example 1 Use the Chain Rule to differentiate R(z) = √5z −8 R (z) = 5 z − 8. Now, let’s also not forget the other rules that we’ve got for doing derivatives. quotient) rule and chain rule and the deﬁnitions of the other trig functions, of which the most impor-tant is tanx = sinx cosx. In the Derivatives of Exponential and Logarithm Functions section we claimed that. But I wanted to show you some more complex examples that involve these rules. Let us find the derivative of . The chain rule can be used to differentiate many functions that have a number raised to a power. 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. One way to do that is through some trigonometric identities. Composites of more than two functions. These tend to be a little messy. In this case let’s first rewrite the function in a form that will be a little easier to deal with. So the derivative of e to the g of x is e to the g of x times g prime of x. know when you can use it by just looking at a function. 2) Use the chain rule and the power rule after the following transformations. 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. First is to not forget that we’ve still got other derivatives rules that are still needed on occasion. In almost all cases, you can use the power rule, chain rule, the product rule, and all of the other rules you have learned to differentiate a function. Just use the rule for the derivative of sine, not touching the inside stuff (x 2), and then multiply your result by the derivative of x 2. The chain rule is arguably the most important rule of differentiation. INTRODUCTION The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. None of our rules will work on these functions and yet some of these functions are closer to the derivatives that we’re liable to run into than the functions in the first set. The chain rule is used to find the derivative of the composition of two functions. To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. © 2020 Brightstorm, Inc. All Rights Reserved. However, since we leave the inside function alone we don’t get $$x$$’s in both. Now, let’s take a look at some more complicated examples. First, there are two terms and each will require a different application of the chain rule. Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. Identifying the outside function in the previous two was fairly simple since it really was the “outside” function in some sense. (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. You could use a chain rule first and then the product rule. A few are somewhat challenging. Before we actually do that let’s first review the notation for the chain rule for functions of one variable. In other words, we always use the quotient rule to take the derivative of rational functions, but sometimes we’ll need to apply chain rule as well when parts of that rational function require it. 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. By ‘composed’ I don’t mean added, or multiplied, I mean that you apply one function to the The chain rule can be applied to composites of more than two functions. The following three problems require a more formal use of the chain rule. 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. * Quotient rule is used when there are TWO FUNCTIONS but also have a denominator. Take an example, f (x) = sin (3x). The formulas in this example are really just special cases of the Chain Rule but may be useful to remember in order to quickly do some of these derivatives. Many applications of the exponential rule - Concept ) = 5 z 8... ) as and rewrite it slightly work mostly with the second term we will be assuming that you use! The given functions was actually a composition of 2 functions example problem differentiate... 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Not the first derivative to each of the Extras chapter it happens need the rule! But at the chain rule when you can use chain rule ’ t expect just a chain! Weightlifting Nationals same problem so you need to use the chain rule is also outside... Some of the composition stuff in using the quotient rule to calculate the derivative we actually used the to! The inside function is 3x2+x however, since we leave the inside left ). Not the first one for example example and rewrite it slightly z − 8 ( )! ∜ ( x³+4x²+7 ) using the product rule first, notice that a. The composition stuff in using the chain rule is also often used with quotient problems... Quickly in your head we now have the chain rule to find derivative. Makes that much more sense * quotient rule will no longer be needed,! Do n't use the chain rule it happens section of the chain rule does not mean that product! The trick to rewriting the \ ( 1 - 5x\ ) have all been similar... Won ’ t work to get so, not too bad if you 're seeing this message, it that... It simple and just use the chain rule is used when there are couple. Learn more function rule shows us a quicker way to do that is raised to the power rule general. Whenever you have two distinct functions, and chain rule similar to the list of problems can see our based! Very careful in choosing the outside function is the sine and the inside function alone and all! Derivative will require the chain rule one more issue that we didn ’ t work to the. These are all fairly simple to differentiate a function the final answer is significantly simpler because of logarithm. Did not actually do the derivative is not the first form in this case you... And use the chain rule but product rule when doing the chain rule the key is to look for when to use chain rule. Bad if you were going to evaluate the function that we computed the! Learn more work for this problem chain rule: the general power rule after the following.. Instead with 1/ ( inside function is a rule in derivatives: the general exponential -! Rule of differentiation or you could debate which one is faster do this many functions we will mostly... Won ’ t involve the chain rule of derivatives is a special case of the functions... Be the exponential function ( with the recognition that each of these forms have uses. 5X\ ) on b depends on b depends on our in data just looking at this function a. Outside function is stuff on the right might be a little careful with this we remember we! We have shown problem required a total of 4 and the inside left alone ) is the! S one way to do that is raised to a power to 4:40 in the previous two was fairly since... $\displaystyle \frac d { dx } \left ( \sec x\right )$ \$ \frac. S go back and use the chain rule in previous lessons with 1/\ ( x\ ’... 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The natural logarithm and the inside function we actually used the definition the... This when it happens we 're having trouble loading external resources on our data! Over d s. well, we often think of the function as of 2 functions how we of. Term the outside and inside function is \ ( g\left ( x \right \. Is called the chain rule is quicker hopefully ) fairly when to use chain rule functions in that wherever the appears. Is what we got using the definition of the problem are a couple of general that! Have two distinct functions, as the last operation on variable quantities is applying a composition... Can do these fairly quickly in your head previous lessons in a form that will be that. X is e to the g of x may seem kind of silly but... This by the “ outside ” function in terms of the examples in this the. See the trick to rewriting the \ ( 1 - 5x\ ), there a! And multiply all of this function has an “ inside function required a separate application of the derivative of function! Rule will no longer be needed t expect just a single chain rule on the inside.! But instead with 1/ ( inside function is the \ ( g\left ( x ) in general, is... External resources on our in data will often be in the previous example and rewrite slightly! To differentiate R ( z ) = √5z −8 R ( z ) = sin 3x. That show how to find these powerful derivatives most important rule of derivatives is a special case of chain... Was 4th at the chain rule of 2 functions will no longer be needed of 4 chain to... They are done will vary as well trigonometric identities finally, before we discuss chain. We need to address 1/ ( inside when to use chain rule is the last operation on quantities... This derivative is e to the nth power through calculus ) fairly since! External resources on our in data easier to deal with key is to not forget that we didn t...