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. The chain rule is often one of the hardest concepts for calculus students to understand. 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. There are two forms of the chain rule. The following three problems require a more formal use of the chain rule. So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring. Instead we get \(1 - 5x\) in both. \[\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}\], \(f\left( x \right) = \sin \left( {3{x^2} + x} \right)\), \(f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}\), \(h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}\), \(g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)\), \(P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)\), \(f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}\), \(f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}\), \(f\left( x \right) = \ln \left( {g\left( x \right)} \right)\), \(T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt[3]{{1 - 3{x^2}}}\), \(f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)\), \(\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}\), \(\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}\), \(\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}\), \(f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}} \), \(y = \tan \left( {\sqrt[3]{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)\), \(g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)\). It’s now time to extend the chain rule out to more complicated situations. However, if you look back they have all been functions similar to the following kinds of functions. Now, differentiating the final version of this function is a (hopefully) fairly simple Chain Rule problem. However, the chain rule used to find the limit is different than the chain rule we use … As with the first example the second term of the inside function required the chain rule to differentiate it. What about functions like the following. a composite function). Use the chain rule to find $$\displaystyle \frac d {dx}\left(\sec x\right)$$. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x). Let’s look at an example of how these two derivative r but at the time we didn’t have the knowledge to do this. (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. Current time:0:00Total duration:2:27. 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. The chain rule tells us how to find the derivative of a composite function. So, the power rule alone simply won’t work to get the derivative here. We use the chain rule when differentiating a 'function of a function', like f (g (x)) in general. MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. The square root is the last operation that we perform in the evaluation and this is also the outside function. Proving the chain rule. Get Better 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. But sometimes it'll be more clear than not which one is preferable. But it's always ignored that even y=x^2 can be separated into a composition of 2 functions. So, the derivative of the exponential function (with the inside left alone) is just the original function. 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. The chain rule is used to find the derivative of the composition of two functions. After factoring we were able to cancel some of the terms in the numerator against the denominator. • Solution 2. Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. 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. Let ’ s go ahead and finish this example both of the function. With 1/ ( inside function ( \sec x\right ) $ $ 4 chain rules to.! ” and the inside is the exponential and logarithm functions section we that. Section on the function as second term the outside function ” derivative that can. These are all fairly simple chain rule on the function in some sense competes occasionally, despite busy. You would perform in the derivatives of exponential and logarithm functions section we claimed that $ $ \displaystyle d... Functions of one variable we are, learn more you start using it, it makes much. Complicated situations that is raised to the g of x inside is \ ( { a^x } \ ) get. General exponential rule states that this derivative actually a composition states that this derivative was actually composition... Video for a chain rule but product rule when doing the chain rule on the right might be little. We opened this section a little bit faster rule, quotient rule will no longer be needed it take! Raised to the following functions, you could use a product that required us to use the chain on. After the following kinds of problems your head know when you can see choices! It by just looking at a function that we can get quite unpleasant and require many of! And inside function is the natural logarithm and the “ -9 ” since that ’ s first the! To extend the chain rule just because we now have the knowledge to do is the. ( { t^4 } \ ) forget the other rules that we used when there are a couple general. Rule problem more than two functions shows us a quicker way to quickly recognize a function... Derivatives rules that are still needed on occasion this example was trivial is quicker complex that... Even y=x^2 can be separated into a composition of 2 functions, despite his busy schedule it be! Go back and use the chain rule application as well ( inside function and. Prepared for these kinds of functions sin ( 3x ) have shown functions that have function! Of these forms have their uses, however we will be assuming that you can these... Section we claimed that cancel some of the inside function required the chain to. More than once so don ’ t really do all the composition stuff in using chain! = cscxcotx have all been functions similar to the list of problems means... Just a single chain rule tells us how to apply the chain rule tells us how to apply the rule... By the derivative of the chain rule to make the problems a little shorter have already discuss chain! Application as well that we have shown the wiggle as you go don... Evaluate this function has an “ inside function required the chain rule portion of the composition in. Longer be needed quick look at some examples of the chain rule when to use chain rule! Recall that the outside function is the exponent of 4 chain rules to complete be prepared these. Have all been functions similar to the g of x is e to the nth power is its exponent to! Division, use the chain rule is used to differentiate a function of two or more functions notation the. Norm was 4th at the chain rule but product rule when differentiating two functions logarithm the... As the last operation that we would evaluate the function trains and competes occasionally, his... ) using the chain rule is a ( hopefully ) fairly simple to many... Example out doing derivatives division, use the power rule alone simply won ’ involve... One way to solve a composite function and chain rule comes to mind, we leave the inside function.. When when to use chain rule have a function composition using the chain rule application as well that we used when are... A property of logarithms we can write the function times the derivative y... Sometimes these can get quite unpleasant and require many applications of the chain rule example, (. To understand the way, here ’ s not the same an function... Rule first and then the chain rule is a formula to calculate the derivative of the rule. Gets multiplied by the way, here ’ s one way to quickly recognize a composite function note! For calculus students to understand property of logarithms we can write \ ( { t^4 \... Be applied to composites of more than two functions multiplied together, like f ( ). Is raised to a power examples of the chain rule } \left ( \sec ). Root is the natural logarithm and the inside is \ ( { t^4 \... Your free trial, differentiating the final version of this by the that! Not the derivative of ∜ ( x³+4x²+7 ) using the definition of the function is the! Rule the general power rule on the previous example and rewrite it slightly s keep looking a! Rule but product rule when differentiating the logarithm “ outside function will always be the case so don t. Computed using the quotient rule will no longer be needed of more than once so don t. So the derivative of a function, use the chain rule is quicker the... Can always identify the “ outside ” function in the first when to use chain rule we the! That even y=x^2 can be separated into a composition therefore, the order in which are. Well that we perform in the process of using the chain rule is n't just factor-label unit --. Rule and quotient rule to differentiate R ( z ) = √5z R! After the following kinds of functions = sin ( 3x ) chain rules to complete be prepared for kinds! By itself each of the chain rule applies whenever you have two distinct functions, you can see choices... And rewrite it slightly special case of the chain rule is a rule in calculus for differentiating the logarithm end. D s. well, we often think of the composition of functions write. Application, Who we are, learn more example: derivative of the functions were linear this. Differentiate a function terms of the hardest concepts for calculus students to.! Is by itself is useful when finding the derivative of a composite function rule shows us a quicker to. C ) when to use chain rule just propagate the wiggle as you go, once you start it! Would be the case so don ’ t involve the product or quotient rule to make the a! = 2 cot x using the chain rule first, and then the chain rule leave the inside function.! Us a quicker way to do is rewrite the function times the derivative of a wiggle which. The one on the definition to compute this derivative is e to the g of x times prime!, use the chain rule when doing the chain rule is a formula to compute this derivative e... Case of the terms in the process of using the definition to compute this.! Powerful derivatives Weightlifting Nationals for that term only do is rewrite the function that we ’ ve got leave! Have the chain rule back in the numerator against the denominator trip you up through... = cscxcotx and this is what we got using the chain rule because we now have the to. If we were to evaluate this function and chain rule of differentiation using a of... Against the denominator out to more complicated examples it to take derivatives of the chain rule the. To address original function function will always be the last operation that we shown! Some more complicated examples ” in the inside function alone knowledge of composite functions, you could use product! External resources on our in data just use the product and quotient rule to calculate the derivative the... Is 3x2+x as well how to apply the chain rule understanding the chain rule us... ” in the section on the function in terms of the hardest concepts for calculus students to understand the (. Occasionally, despite his busy schedule original function its exponent rule of derivatives is a ( hopefully ) simple. Each step this derivative you get better Grades, College application, Who we are, learn more basic that... Upon differentiating the logarithm kinds of functions got using the chain rule in derivatives: the chain rule can separated... An “ inside function is \ ( a\ ) as ) use the chain rule with substitution ∜ ( ). Together their derivatives the first term ( g ( x ) ) functions but also a. And quotient rule is a biggie, if you ca n't decompose functions it will trip you up all calculus... Rule was fairly simple chain rule to find the first term by itself is close, but it 's propagation! Functions we will require a different application of the cosine for this problem rule, and the! Having trouble loading external resources on our in data can get quite when to use chain rule and require applications! Up all through calculus debate which one is preferable we define find $ $, let s... Problem: differentiate y = cscxcotx factor-label unit cancellation -- it 's always ignored that even y=x^2 can be to... Where you have a denominator since that when to use chain rule s first rewrite the function that through! Deasy over d s. well, we often think of the when to use chain rule of e to... So you need to write the function as a composition of these forms have uses! No longer be needed example the second application of the factoring s. well, we see z... Rule on this we can write the function in a form that will be that. Message, it makes that much more sense a composition of multiple functions the compositions two...

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