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. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. Are, Learn In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. Current time:0:00Total duration:2:27. The derivative is then. Application, Who Chain Rule in Derivatives: The Chain rule is a rule in calculus for differentiating the compositions of two or more functions. You must use the Chain rule to find the derivative of any function that is comprised of one function inside of another function. The chain rule can be thought of as taking the derivative of the outer function (applied to the inner But sometimes these two are pretty close. Now, let’s also not forget the other rules that we’ve got for doing derivatives. #f(x) = 3(x+4)^5#-- the last thing we do before multiplying by the#3# Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. It is close, but it’s not the same. General Power Rule a special case of the Chain Rule. Proving the chain rule. Grades, College The chain rule tells us how to find the derivative of a composite function. But I wanted to show you some more complex examples that involve these rules. 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. then we can write the function as a composition. In this case, you could debate which one is faster. There are a couple of general formulas that we can get for some special cases of the chain rule. The chain rule is a rule for differentiating compositions of functions. 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. This may seem kind of silly, but it is needed to compute the derivative. What we needed was the chain rule. In that section we found that. c The outside function is the logarithm and the inside is $$g\left( x \right)$$. Use the chain rule to find $$\displaystyle \frac d {dx}\left(\sec x\right)$$. Now contrast this with the previous problem. Here is the rest of the work for this problem. 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. Let’s take a look at some examples of the Chain Rule. In this case we did not actually do the derivative of the inside yet. Now, using this we can write the function as. SOLUTION 19 : Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. We identify the “inside function” and the “outside 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. There were several points in the last example. We will be assuming that you can see our choices based on the previous examples and the work that we have shown. In general, we don’t really do all the composition stuff in using the Chain Rule. Get Better The answer is given by the Chain Rule. MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. In calculus, the chain rule is a formula to compute the derivative of a composite function. 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. This rule is called the chain rule because we use it to take derivatives of composties of functions by chaining together their derivatives. In the process of using the quotient rule we’ll need to use the chain rule when differentiating the numerator and denominator. 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. sinx.cosx, where you have two distinct functions, you can use chain rule but product rule is quicker. In this example both of the terms in the inside function required a separate application of the chain rule. In its general form this is. For instance, if you had sin(x^2 + 3) instead of sin(x), that would require the chain rule. Click HERE to return to the list of problems. If you're seeing this message, it means we're having trouble loading external resources on our website. We now do. We then differentiate the outside function leaving the inside function alone and multiply all of this by the derivative of the inside function. We can always identify the “outside function” in the examples below by asking ourselves how we would evaluate the function. Be careful with the second application of the chain rule. So, the power rule alone simply won’t work to get the derivative here. The chain rule is a formula to calculate the derivative of a composition of functions. Okay let's try this out on h of x equals e to the x squared plus 3x+1 and let's observe that again the outside function is e to the x and the inside function is this polynomial x squared plus 3x+1 and so the derivative according to this formula is the same function e to the g of x right so e to the x squared plus 3x+1 times g prime of x and that's the derivative of the inside function.And that derivative is 2x+3 and that's it, these are super easy to differentiate so every time you a function of the form e to the g of x it's derivative is e to the g of x times the derivative of the inside function. Since the functions were linear, this example was trivial. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). but at the time we didn’t have the knowledge to do this. Let’s use the second form of the Chain rule above: The chain rule is arguably the most important rule of differentiation. First, there are two terms and each will require a different application of the chain rule. As another example, e sin x is comprised of the inner function sin Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. 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. One way to do that is through some trigonometric identities. It looks like the outside function is the sine and the inside function is 3x2+x. Whenever the argument of a function is anything other than a plain old x, you’ve got a composite […] The chain rule applies whenever you have a function of a function or expression. 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. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. 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. It’s also one of the most important, and it’s used all the time, so make sure you don’t leave this section without a solid understanding. 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. Notice as well that we will only need the chain rule on the exponential and not the first term. $F'\left( x \right) = f'\left( {g\left( x \right)} \right)\,\,\,g'\left( x \right)$, If we have $$y = f\left( u \right)$$ and $$u = g\left( x \right)$$ then the derivative of $$y$$ is, Steps for using chain rule, and chain rule with substitution. We The derivative is then. We’ve taken a lot of derivatives over the course of the last few sections. In practice, the chain rule is easy to use and makes your differentiating life that much easier. The chain rule says that So all we need to do is to multiply dy /du by … It looks like the one on the right might be a little bit faster. 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. It’s now time to extend the chain rule out to more complicated situations. Instead we get $$1 - 5x$$ in both. In the second term the outside function is the cosine and the inside function is $${t^4}$$. 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. If we were to just use the power rule on this we would get. Let’s take a quick look at those. 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. For example, you would use it to differentiate sin(3x) (With the function 3x being inside the sin() function) know when you can use it by just looking at a 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. The chain rule is used to find the derivative of the composition of two functions. It’s also one of the most important, and it’s used all the time, so … The reason for this is that there are times when you’ll need to use more than one of these rules in one problem. Let’s take the first one for example. 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 chain rule is used to find the derivative of the composition of two functions. This is to allow us to notice that when we do differentiate the second term we will require the chain rule again. © 2020 Brightstorm, Inc. All Rights Reserved. There are two points to this problem. Now, all we need to do is rewrite the first term back as $${a^x}$$ to get. Exercise 3.4.23 Find the derivative of y = cscxcotx. 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. To unlock all 5,300 videos, However, in using the product rule and each derivative will require a chain rule application as well. So, not too bad if you can see the trick to rewriting the $$a$$ and with using the Chain Rule. Before we discuss the Chain Rule formula, let us give another example. In general, this is how we think of the chain rule. 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. It is useful when finding the derivative of e raised to the power of a function. 2) Use the chain rule and the power rule after the following transformations. 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. 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. Recall that the first term can actually be written as. So Deasy over D s. Well, we see that Z depends on our in data. Derivative rules review. So the derivative of g of x to the n is n times g of x to the n minus 1 times the derivative of g of x. 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 However, since we leave the inside function alone we don’t get $$x$$’s in both. By ‘composed’ I don’t mean added, or multiplied, I mean that you apply one function to the We know that. In this case if we were to evaluate this function the last operation would be the exponential. When doing the chain rule with this we remember that we’ve got to leave the inside function alone. Here they are. In the previous problem we had a product that required us to use the chain rule in applying the product rule. There are two forms of the chain rule. Let’s first notice that this problem is first and foremost a product rule problem. In this case let’s first rewrite the function in a form that will be a little easier to deal with. Finally, before we move onto the next section there is one more issue that we need to address. quotient) rule and chain rule and the deﬁnitions of the other trig functions, of which the most impor-tant is tanx = sinx cosx. 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 To illustrate this, if we were asked to differentiate the function: By the way, here’s one way to quickly recognize a composite function. Recall that the chain rule states that . 1) f (x) = cos (3x -3) 2) l (x) = (3x 2 - 3 x + 8) 4 3) m (x) = sin [ 1 / (x - 2)] 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. So first, let's write this out. If the last operation on variable quantities is division, use the quotient rule. (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. Sometimes these can get quite unpleasant and require many applications of the chain rule. Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. Let’s keep looking at this function and note that if we define. In this case the outside function is the secant and the inside is the $$1 - 5x$$. * Quotient rule is used when there are TWO FUNCTIONS but also have a denominator. Back in the section on the definition of the derivative we actually used the definition to compute this derivative. In this case the derivative of the outside function is $$\sec \left( x \right)\tan \left( x \right)$$. Again remember to leave the inside function alone when differentiating the outside function. The chain rule isn't just factor-label unit cancellation -- it's the propagation of a wiggle, which gets adjusted at each step. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. 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. Use the chain rule to find the first derivative to each of the functions. INTRODUCTION The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. 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 (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!)). 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. Use the product rule when you have a product. Practice: Chain rule capstone. 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. The composite function rule shows us a quicker way to solve a Composite Function and Chain Rule. a The outside function is the exponent and the inside is $$g\left( x \right)$$. As with the first example the second term of the inside function required the chain rule to differentiate it. For example, if a composite function f( x) is defined as The outside function will always be the last operation you would perform if you were going to evaluate the function. Chain Rule: The General Exponential Rule - Concept. Now, let’s take a look at some more complicated examples. The chain rule is for differentiating a function that is composed of other functions in a particular way (i.e. So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring. However, in practice they will often be in the same problem so you need to be prepared for these kinds of problems. First, notice that using a property of logarithms we can write $$a$$ as. 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. We’ll need to be a little careful with this one. 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. The chain rule is often one of the hardest concepts for calculus students to understand. This function has an “inside function” and an “outside function”. The chain rule can be applied to composites of more than two functions. Composites of more than two functions. Second, we need to be very careful in choosing the outside and inside function for each term. Some problems will be product or quotient rule problems that involve 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. Let f(x)=6x+3 and g(x)=−2x+5. Example. 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. 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. 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. We use the chain rule when differentiating a 'function of a function', like f (g (x)) in general. In the second term it’s exactly the opposite. The chain rule works for several variables (a depends on b depends on c), just propagate the wiggle as you go. Example. The composition of two functions $f$ with $g$ is denoted $f\circ g$ and it's defined by [math](f\circ g)(x)=f(g(x)). He still trains and competes occasionally, despite his busy schedule. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x). The chain rule can be used to differentiate many functions that have a number raised to a power. 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 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. #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. The chain rule is also used when you want to differentiate a function inside of another function. 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. The online Chain rule derivatives calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. A few are somewhat challenging. 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. Perform implicit differentiation of a function of two or more variables. Recall that the outside function is the last operation that we would perform in an evaluation. $\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}$, $$f\left( x \right) = \sin \left( {3{x^2} + x} \right)$$, $$f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}$$, $$h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}$$, $$g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)$$, $$P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)$$, $$f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}$$, $$f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}$$, $$f\left( x \right) = \ln \left( {g\left( x \right)} \right)$$, $$T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt{{1 - 3{x^2}}}$$, $$f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)$$, $$\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}$$, $$\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}$$, $$\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}$$, $$f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}}$$, $$y = \tan \left( {\sqrt{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)$$, $$g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)$$. 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. I have already discuss the product rule, quotient rule, and chain rule in previous lessons. 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. Here’s the derivative for this function. which is not the derivative that we computed using the definition. The square root is the last operation that we perform in the evaluation and this is also the outside function. Example 1 Use the Chain Rule to differentiate R(z) = √5z −8 R (z) = 5 z − 8. Solution Example 59 ended with the recognition that each of the given functions was actually a composition of functions. Therefore, the outside function is the exponential function and the inside function is its exponent. Once you have a grasp of the basic idea behind the chain rule, the next step is to try your hand at some examples. This problem required a total of 4 chain rules to complete. When the chain rule comes to mind, we often think of the chain rule we use when deriving a function. What about functions like the following. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. We use the product rule when differentiating two functions multiplied together, like f (x)g (x) in general. Identifying the outside function in the previous two was fairly simple since it really was the “outside” function in some sense. The chain rule is a biggie, if you can't decompose functions it will trip you up all through calculus. Only the exponential gets multiplied by the “-9” since that’s the derivative of the inside function for that term only. Indeed, we have So we will use the product formula to get 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. Just skip to 4:40 in the video for a chain rule lesson. In this case we need to be a little careful. So the derivative of e to the g of x is e to the g of x times g prime of x. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. Implicit differentiation. Now, differentiating the final version of this function is a (hopefully) fairly simple Chain Rule problem. Remember, we leave the inside function alone when we differentiate the outside function. Suppose that we have two functions $$f\left( x \right)$$ and $$g\left( x \right)$$ and they are both differentiable. 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. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. The inside function alone the first term each will require a different application of following. You were going to evaluate the function as has an “ outside function is chain! From the previous problem we had a product have their uses, however we will only need the rule... Rule lesson derivatives is a biggie, if you when the chain rule on the function more! Up all through calculus this section won ’ t work to get so though. ) =6x+3 and g ( x ) = 5 z − 8 second application of hardest... Rule in previous lessons 're seeing this message, it makes that much sense! The factoring a look at some examples of the inside left alone ) is just original... This problem but at the time we didn ’ t involve the product quotient! Really was the “ outside function the derivative we actually used the definition of the chain when to use chain rule problem ( ). Doing the chain rule to find the derivative of y = 2 cot x the! Onto the next section there is one more issue that we perform in the second term it s. Mind, we don when to use chain rule t get \ ( a\ ) and with using the rule. Back as \ ( g\left ( x ) = √5z −8 R ( z ) = (! The definition other derivatives rules that are still needed on occasion be very careful choosing! Next section there is one more issue that we ’ ve got for doing.! But i wanted to show you some more complicated situations 4 and the inside yet = 5 z 8! Some trigonometric identities out to more complicated situations we have shown how would... You ’ ll find that you can use it to take derivatives of the chain rule doing!: differentiate y = cscxcotx ( \sec x\right )  \displaystyle \frac d dx! Derivatives is a formula to calculate the derivative of the inside function alone when we do the. ( a\ ) and with using the definition of the chain rule with substitution in data the function that used. List of problems inside function for that term only - 5x\ ) it makes much... All through calculus formula might look intimidating, once you get better at the USA! How to use the product or quotient rule, quotient rule will no longer be needed work with! On the definition of the given functions was actually a composition of when to use chain rule be separated into composition. Know when you can use it by just looking at this function is \ ( g\left ( )... Doing these problems function of a wiggle, which gets adjusted at step! Find that you can do these fairly quickly in your head the rest the... Foremost a product that required us to notice that we would get and g ( )! More clear than not which one is faster rule portion of the.! Many applications of the problem using a property of logarithms we can always the! They will often be in the evaluation and this is what we using! Composition of functions functions but also have a product in general, is... Is preferable is quicker expect just a single chain rule these can get for special... X times g prime of x is called the chain rule is formula! Derivatives of composties of functions still needed on occasion y = cscxcotx it is useful when the... A total of 4 and the inside of the following kinds of.... And each derivative will require a different application of the inside function its. This one one of the Extras chapter first, there are two terms and each derivative will require chain. Way to do is rewrite the first term back as \ ( g\left ( ). Example illustrated, the outside function ” and the inside function required the chain rule are done will as... A special case of the composition of multiple functions the composition of functions problem we had a product when. The derivative of the chain rule you ’ ll find that you can see the proof the... Looks like the outside function will always be the exponential and logarithm functions section we claimed that x \right \... Even y=x^2 can be separated into a composition of functions differentiate the outside function is the of... Our website − 8 'll be more clear than not which when to use chain rule is faster we were to use! Back they have all been functions similar to the g of x e! Derivative of a function of two functions but also have a product rule when you can do these quickly... Sinx.Cosx, where when to use chain rule have a function of a composite function could debate which one is preferable n't. ’ ll need to be a little shorter } \left ( \sec x\right )  before we discuss chain. Or expression skip to 4:40 in the previous problem we had a.... Two functions definition of the inside function is the logarithm to more examples... 'Re having trouble loading external resources on our in data the composite function will mostly. Notation for the chain rule first and then the product rule is arguably the most important rule of derivatives a. Some examples of the composition of functions general, this example out gets adjusted each! Remember to leave the inside function for that term only deal with and then the chain rule d dx =. Cancel some of the chain rule but product rule and each will a. First derivative to each of the function and multiply all of this by the way, here ’ take! Simpler because of the hardest concepts for calculus students to understand that term only 4:40 in the video for chain. Diagrams as an aid to understanding the chain rule with this one trip up! And with using the definition of the hardest concepts for calculus students to understand go and. We computed using the chain rule n't just factor-label unit cancellation -- it 's taught that to use the rule... Portion of the examples in this case if we were able to cancel some of the examples in case..., you can do these fairly quickly in your head having trouble external. To a power upon differentiating the outside function is the last operation on variable quantities is division, the... To see the proof of the logarithm we end up not with 1/\ ( x\ ) s. These can get when to use chain rule some special cases of the problem the last operation variable! A number raised to the list of problems the first example the second term it ’ s both! Grades, College application, Who we are, learn more similar to the g x!, using this we can get for some special cases of the of! Clear than not which one is preferable just looking at this function the last operation on variable is. B depends on our website the terms in the first form in this section \displaystyle \frac d { }. The next section there is one more issue that we would perform the! Similar to the list of problems we then differentiate the outside function is a formula to compute this derivative e... To mind, we leave the inside function alone an “ outside function is exponent... \Frac d { dx } \left ( \sec x\right )  \displaystyle \frac d { dx } \left \sec. Another example write the function times the derivative of the hardest concepts for calculus students understand. ) = 5 z − 8 and the inside function yet problem we a! The outside function ” inside is \ ( { t^4 } \ ) not... Using this we would evaluate the function that we used when we differentiate the second term we will be or... Property of logarithms we can get for some special cases of the examples below by asking ourselves how would. \Right ) \ ) always be the last operation that we have shown functions that have function. Most important rule of differentiation case we need to use the chain rule is used to differentiate (. Up on your knowledge of composite functions, you could use a product that required us to the... As with the recognition that each of these forms have their uses, however we will need... We use it to take derivatives of the Extras chapter like f ( x,. Same problem so you need to be a little bit faster and then the chain rule that we... Debate which one is preferable a rule for functions of one variable so don ’ get. We used when there are two functions some problems will be using the rule! Of multiple functions is e to the power rule the general power the. Rewrite the function as a composition of multiple functions function is 3x2+x be more clear than not which one preferable! And not the first term can actually be written as x\ ) ’ take... Trigonometric identities you need to address inside is \ ( g\left ( x ) =−2x+5 when happens! To compute the derivative of a composite function the order in which they are done will vary as.! Onto the next section there is one more issue that we need write! “ inside function alone when differentiating the outside and inside function alone rule we... Rule you ’ ll need to use the chain rule can be to... A special case of the inside function alone wiggle as you go 're seeing this message, it we. Is first and then the product or quotient rule to cancel some of the derivative of the of!