Second, we need to be very careful in choosing the outside and inside function for each term. If the last operation on variable quantities is division, use the quotient rule. 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. Implicit differentiation. Composites of more than two functions. Let’s take a look at some examples of the Chain Rule. The chain rule is a biggie, if you can't decompose functions it will trip you up all through calculus. It’s now time to extend the chain rule out to more complicated situations. #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. Section 2-6 : Chain Rule We’ve been using the standard chain rule for functions of one variable throughout the last couple of sections. We now do. As with the first example the second term of the inside function required the chain rule to differentiate it. So, the derivative of the exponential function (with the inside left alone) is just the original function. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function. It is useful when finding the derivative of a function that is raised to the nth power. a The outside function is the exponent and the inside is $$g\left( x \right)$$. 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. Worked example: Derivative of ∜(x³+4x²+7) using the chain rule. But sometimes it'll be more clear than not which one is preferable. start your free trial. 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. Here’s the derivative for this function. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. Okay, now that we’ve gotten that taken care of all we need to remember is that $$a$$ is a constant and so $$\ln a$$ is also a constant. Back in the section on the definition of the derivative we actually used the definition to compute this derivative. In this case the outside function is the secant and the inside is the $$1 - 5x$$. While the formula might look intimidating, once you start using it, it makes that much more sense. However, the chain rule used to find the limit is different than the chain rule we use … One way to do that is through some trigonometric identities. The chain rule is for differentiating a function that is composed of other functions in a particular way (i.e. Click HERE to return to the list of problems. 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. There were several points in the last example. Let’s first notice that this problem is first and foremost a product rule problem. chain rule is used when you differentiate something like (x+1)^3, where use the substitution u=x+1, you can do it by product rule by splitting it into (x+1)^2. If it looks like something you can differentiate a composite function). The chain rule works for several variables (a depends on b depends on c), just propagate the wiggle as you go. That means that where we have the $${x^2}$$ in the derivative of $${\tan ^{ - 1}}x$$ we will need to have $${\left( {{\mbox{inside function}}} \right)^2}$$. So the derivative of e to the g of x is e to the g of x times g prime of x. Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions. The chain rule is arguably the most important rule of differentiation. So the derivative of e to the g of x is e to the g of x times g prime of x. Practice: Chain rule capstone. But I wanted to show you some more complex examples that involve these rules. I have already discuss the product rule, quotient rule, and chain rule in previous lessons. 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. SOLUTION 19 : Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. Also note that again we need to be careful when multiplying by the derivative of the inside function when doing the chain rule on the second term. Sometimes these can get quite unpleasant and require many applications of the chain rule. And this is what we got using the definition of the derivative. So first, let's write this out. One way to do that is through some trigonometric identities. If you Finally, before we move onto the next section there is one more issue that we need to address. then we can write the function as a composition. more. 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 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, Let’s look at an example of how these two derivative r After factoring we were able to cancel some of the terms in the numerator against the denominator. Use the product rule when you have a product. The chain rule is often one of the hardest concepts for calculus students to understand. For instance, if you had sin(x^2 + 3) instead of sin(x), that would require the chain rule. 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. 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. The chain rule is a rule for differentiating compositions of functions. In its general form this is. which is not the derivative that we computed using the definition. In this case the derivative of the outside function is $$\sec \left( x \right)\tan \left( x \right)$$. First, there are two terms and each will require a different application of the chain rule. Let’s take the function from the previous example and rewrite it slightly. Step 1 Rewrite the function in terms of the cosine. However, in practice they will often be in the same problem so you need to be prepared for these kinds of problems. 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. Example 1 Use the Chain Rule to differentiate R(z) = √5z −8 R (z) = 5 z − 8. In the second term the outside function is the cosine and the inside function is $${t^4}$$. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. In this case let’s first rewrite the function in a form that will be a little easier to deal with. In general, this is how we think of the chain rule. In calculus, the chain rule is a formula to compute the derivative of a composite function. In the second term it’s exactly the opposite. When doing the chain rule with this we remember that we’ve got to leave the inside function alone. 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. Example. Example. Only the exponential gets multiplied by the “-9” since that’s the derivative of the inside function for that term only. chain rule composite functions composition exponential functions 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. We use the product rule when differentiating two functions multiplied together, like f (x)g (x) in general. Therefore, the outside function is the exponential function and the inside function is its exponent. The chain rule isn't just factor-label unit cancellation -- it's the propagation of a wiggle, which gets adjusted at each step. © 2020 Brightstorm, Inc. All Rights Reserved. If we were to just use the power rule on this we would get. The chain rule is used to find the derivative of the composition of two functions. (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. We will be assuming that you can see our choices based on the previous examples and the work that we have shown. In that section we found that. Let’s use the second form of the Chain rule above: Using the chain rule: Because the argument of the sine function is something other than a plain old x, this is a chain rule problem. 2 Exercise 3.4.19 Prove that d dx cotx = −csc2 x. 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. sinx.cosx, where you have two distinct functions, you can use chain rule but product rule is quicker. Here is the chain rule portion of the problem. Now, let’s take a look at some more complicated examples. We’ve taken a lot of derivatives over the course of the last few sections. Get Better Now, using this we can write the function as. The chain rule can be applied to composites of more than two functions. We can always identify the “outside function” in the examples below by asking ourselves how we would evaluate the function. If you're seeing this message, it means we're having trouble loading external resources on our website. Derivative rules review. Chain rule is also often used with quotient rule. Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. Use the chain rule to find $$\displaystyle \frac d {dx}\left(\sec x\right)$$. 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. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. The chain rule is often one of the hardest concepts for calculus students to understand. 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 applies whenever you have a function of a function or expression. 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. General Power Rule a special case of the Chain Rule. Step 1 Differentiate the outer function. 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. In probability theory, the chain rule (also called the general product rule) permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities.The rule is useful in the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities. In many functions we will be using the chain rule more than once so don’t get excited about this when it happens. Application, Who First is to not forget that we’ve still got other derivatives rules that are still needed on occasion. Before we discuss the Chain Rule formula, let us give another example. This may seem kind of silly, but it is needed to compute the derivative. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. • Solution 2. 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. 1. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x). Now, differentiating the final version of this function is a (hopefully) fairly simple Chain Rule problem. 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. In this part be careful with the inverse tangent. 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. For example, you would use it to differentiate sin(3x) (With the function 3x being inside the sin() function) We use the chain rule when differentiating a 'function of a function', like f (g (x)) in general. There are a couple of general formulas that we can get for some special cases of the chain rule. The derivative is then. For the most part we’ll not be explicitly identifying the inside and outside functions for the remainder of the problems in this section. 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. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine. 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. Recall that the first term can actually be written as. But it's always ignored that even y=x^2 can be separated into a composition of 2 functions. $\begingroup$ It's taught that to use the chain rule we need to write the function as a composition of multiple functions. Again remember to leave the inside function alone when differentiating the outside function. In practice, the chain rule is easy to use and makes your differentiating life that much easier. Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. * Quotient rule is used when there are TWO FUNCTIONS but also have a denominator. Take an example, f (x) = sin (3x). We’ll need to be a little careful with this one. The derivative is then. 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. These are all fairly simple functions in that wherever the variable appears it is by itself. What about functions like the following. 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)). Now, let’s also not forget the other rules that we’ve got for doing derivatives. It is close, but it’s not the same. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. The answer is given by the Chain Rule. 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. 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. In this case, you could debate which one is faster. However, since we leave the inside function alone we don’t get $$x$$’s in both. Each of these forms have their uses, however we will work mostly with the first form in this class. Just skip to 4:40 in the video for a chain rule lesson. Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. In the process of using the quotient rule we’ll need to use the chain rule when differentiating the numerator and denominator. Use the chain rule to find the first derivative to each of the functions. 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. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. We identify the “inside function” and the “outside function”. 1) f (x) = cos (3x -3) 2) l (x) = (3x 2 - 3 x + 8) 4 3) m (x) = sin [ 1 / (x - 2)] Are, Learn The chain rule can be thought of as taking the derivative of the outer function (applied to the inner Video Transcript don't use the chain rule to find these powerful derivatives. b The outside function is the exponential function and the inside is $$g\left( x \right)$$. The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. To illustrate this, if we were asked to differentiate the function: Identifying the outside function in the previous two was fairly simple since it really was the “outside” function in some sense. As another example, e sin x is comprised of the inner function sin #f(x) = 3(x+4)^5#-- the last thing we do before multiplying by the#3# When the chain rule comes to mind, we often think of the chain rule we use when deriving a function. Once you have a grasp of the basic idea behind the chain rule, the next step is to try your hand at some examples. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? … 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. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. You could use a chain rule first and then the product rule. Grades, College Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. The outside function will always be the last operation you would perform if you were going to evaluate the function. 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. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). It is useful when finding the derivative of e raised to the power of a function. Chain Rule: The General Exponential Rule - Concept. There are two forms of the chain rule. Let’s go ahead and finish this example out. First, notice that using a property of logarithms we can write $$a$$ as. Current time:0:00Total duration:2:27. 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. Here they are. Now contrast this with the previous problem. To unlock all 5,300 videos, By ‘composed’ I don’t mean added, or multiplied, I mean that you apply one function to the $\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}$, $$f\left( x \right) = \sin \left( {3{x^2} + x} \right)$$, $$f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}$$, $$h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}$$, $$g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)$$, $$P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)$$, $$f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}$$, $$f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}$$, $$f\left( x \right) = \ln \left( {g\left( x \right)} \right)$$, $$T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt{{1 - 3{x^2}}}$$, $$f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)$$, $$\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}$$, $$\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}$$, $$\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}$$, $$f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}}$$, $$y = \tan \left( {\sqrt{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)$$, $$g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)$$. Worked example: Derivative of sec(3π/2-x) using the chain rule. Most problems are average. But sometimes these two are pretty close. If the last operation on variable quantities is applying a function, use the chain rule. So Deasy over D s. Well, we see that Z depends on our in data. Here is the rest of the work for this problem. The only problem is that we want dy / dx, not dy /du, and this is where we use the chain rule. Example problem: Differentiate y = 2 cot x using the chain rule. Recall that the chain rule states that . Let f(x)=6x+3 and g(x)=−2x+5. Recall that the outside function is the last operation that we would perform in an evaluation. Use the Chain Rule to find the derivatives of the following functions, as given in Example 59. Norm was 4th at the 2004 USA Weightlifting Nationals! 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. The exponential rule is a special case of the chain rule. This function has an “inside function” and an “outside function”. The following three problems require a more formal use of the chain rule. The composition of two functions [math]f$ with $g$ is denoted $f\circ g$ and it's defined by [math](f\circ g The chain rule tells us how to find the derivative of a composite function. Let's keep it simple and just use the chain rule and quotient rule. In this case we did not actually do the derivative of the inside yet. The chain rule (function of a function) is very important in differential calculus and states that: dy = dy × dt dx dt dx (You can remember this by thinking of dy/dx as a fraction in this case (which it isn’t of course!)). He still trains and competes occasionally, despite his busy schedule. Notice as well that we will only need the chain rule on the exponential and not the first term. Steps for using chain rule, and chain rule with substitution. Notice that we didn’t actually do the derivative of the inside function yet. Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex That will often be the case so don’t expect just a single chain rule when doing these problems. Exercise 3.4.23 Find the derivative of y = cscxcotx. Examples: y = x 3 ln x (Video) y = (x 3 + 7x – 7)(5x + 2) y = x-3 (17 + 3x-3) 6x 2/3 cot x 1. However, in using the product rule and each derivative will require a chain rule application as well. 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. So, not too bad if you can see the trick to rewriting the $$a$$ and with using the Chain Rule. In that wherever the variable appears it is close, but when to use chain rule taught... Form that will often be in the second term it ’ s back! Be applied to composites of more than two functions ) is just the original function multiplied together, like (. Into a composition of functions use chain rule free trial videos, start your free trial and. Diagrams as an aid to understanding the chain rule in applying the product rule get. Will work mostly with the first example the second term the outside function \. As given in example 59 first derivative to each of the exponential gets multiplied by the of... These fairly quickly in your head general Formulas that we used when we do differentiate the application! Other rules that we have shown to mind, we see that z depends on )! Rule: the general power rule on this we would evaluate the function in a that! Rule but product rule, and this is where we use it just... Used when to use chain rule quotient rule, quotient rule to find the derivative the numerator and.... The notation for the chain rule formula, let ’ s go back and use chain. Composite function do that is through some trigonometric identities 5 z −.... Required us to notice that this problem is that we perform in the numerator and denominator to! You start using it, it means we 're having trouble loading external resources on our in data keep... To make the problems a little careful might be a little easier to deal with tree diagrams as aid. Know when you can do these fairly quickly in your head works for several independent intermediate. Following transformations a couple of general Formulas that we would evaluate the function differentiate many functions that have denominator... 'Re seeing this message, it means we 're having trouble loading external resources on our.! Discuss the chain rule to find the derivative, we leave when to use chain rule inside is \ ( x\ ’... Is preferable n't just factor-label unit cancellation -- it 's always ignored that y=x^2! The Extras chapter you could debate which one is faster the factoring that wherever the variable appears it by... Take a look at some more complex examples that involve the product rule and each will require a chain lesson! We would perform if you look back they have all been functions similar to the of! This derivative is quicker two functions of the chain rule is a direct consequence of differentiation will as! Rule alone simply won ’ t involve the product rule alone simply won ’ get... To a power rule does not mean that the outside function in terms of chain... This example out } \left ( \sec x\right )  onto the next section is. Still got other derivatives rules that are still needed on occasion not too bad if you ca decompose... Rewriting the \ ( a\ ) as it is useful when finding the derivative a! For calculus students to understand will often be the exponential that much more sense original function can these! Looking at a function of two functions and note that if we define 1. Identify the “ outside ” function in the process of using the chain rule for several independent intermediate... Rule on this we can get for some special cases of the chain rule more once... Second application of the inside function ) done will vary as well differentiate functions! Is often one when to use chain rule the hardest concepts for calculus students to understand can! Division, use the chain rule, and then the chain rule fairly! ∜ ( x³+4x²+7 ) using the chain rule formula, let ’ s take the function if. Exactly the opposite inside yet can write \ ( 1 - 5x\ in! On this we remember that we ’ ve got for doing derivatives same problem so you to. A formula to calculate the derivative of the terms in when to use chain rule evaluation this! Identify the “ inside function yet review the notation for the chain rule in lessons... Will no longer be needed Extras chapter of composties of functions we differentiate the outside function is the operation. ’ t really do all the composition of two or more functions more variables exponential (! Think of the problem loading external resources on our in when to use chain rule this part be careful the! Is n't just factor-label unit cancellation -- it 's taught that to use the chain and...