# when to use chain rule

If the last operation on variable quantities is division, use the quotient rule. Step 1 Differentiate the outer function. Recall that the first term can actually be written as. What about functions like the following. The outside function will always be the last operation you would perform if you were going to evaluate the function. Most problems are average. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine. So Deasy over D s. Well, we see that Z depends on our in data. After factoring we were able to cancel some of the terms in the numerator against the denominator. We identify the “inside function” and the “outside function”. 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. While the formula might look intimidating, once you start using it, it makes that much more sense. Now, using this we can write the function as. 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. In mathematical analysis, the chain rule is a derivation rule that allows to calculate the derivative of the function composed of two derivable functions. Most of the examples in this section won’t involve the product or quotient rule to make the problems a little shorter. Here is the rest of the work for this problem. One way to do that is through some trigonometric identities. In this case we did not actually do the derivative of the inside yet. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ? The chain rule says that So all we need to do is to multiply dy /du by … Use the product rule when you have a product. 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 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)). and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule. 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. The chain rule is a biggie, if you can't decompose functions it will trip you up all through calculus. 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. start your free trial. 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. These are all fairly simple functions in that wherever the variable appears it is by itself. 2) Use the chain rule and the power rule after the following transformations. It’s also one of the most important, and it’s used all the time, so … #f(x) = 3(x+4)^5#-- the last thing we do before multiplying by the#3# 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. Before we discuss the Chain Rule formula, let us give another example. It looks like the outside function is the sine and the inside function is 3x2+x. 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. Example. I have already discuss the product rule, quotient rule, and chain rule in previous lessons. 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. 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. Use the chain rule to find $$\displaystyle \frac d {dx}\left(\sec x\right)$$. which is not the derivative that we computed using the definition. The chain rule can be thought of as taking the derivative of the outer function (applied to the inner So the derivative of e to the g of x is e to the g of x times g prime of x. 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. MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. Proving the chain rule. 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 can be applied to composites of more than two functions. Notice that we didn’t actually do the derivative of the inside function yet. 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. The chain rule tells us how to find the derivative of a composite function. This is to allow us to notice that when we do differentiate the second term we will require the chain rule again. 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. Recall that the chain rule states that . First is to not forget that we’ve still got other derivatives rules that are still needed on occasion. Some problems will be product or quotient rule problems that involve the chain rule. One way to do that is through some trigonometric identities. b The outside function is the exponential function and the inside is $$g\left( x \right)$$. We use the product rule when differentiating two functions multiplied together, like f (x)g (x) in general. The chain rule is arguably the most important rule of differentiation. As with the first example the second term of the inside function required the chain rule to differentiate it. 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. In general, we don’t really do all the composition stuff in using the Chain Rule. What we needed was the chain rule. Now, all we need to do is rewrite the first term back as $${a^x}$$ to get. 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. Let us find the derivative of . Here is the chain rule portion of the problem. The chain rule is used to find the derivative of the composition of two functions. These tend to be a little messy. 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. We will be assuming that you can see our choices based on the previous examples and the work that we have shown. Steps for using chain rule, and chain rule with substitution. If you're seeing this message, it means we're having trouble loading external resources on our website. Application, Who Practice: Chain rule capstone. Identifying the outside function in the previous two was fairly simple since it really was the “outside” function in some sense. 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. The exponential rule states that this derivative is e to the power of the function times the derivative of the function. So the derivative of e to the g of x is e to the g of x times g prime of x. It is close, but it’s not the same. 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. For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. To unlock all 5,300 videos, First, notice that using a property of logarithms we can write $$a$$ as. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. If we were to just use the power rule on this we would get. Chain rule is also often used with quotient rule. © 2020 Brightstorm, Inc. All Rights Reserved. In this case the outside function is the secant and the inside is the $$1 - 5x$$. You could use a chain rule first and then the product rule. 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 you know when you can use it by just looking at a function. In this case, you could debate which one is faster. 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. Implicit differentiation. The exponential rule is a special case of the chain rule. 1) f (x) = cos (3x -3) 2) l (x) = (3x 2 - 3 x + 8) 4 3) m (x) = sin [ 1 / (x - 2)] 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. Before we discuss the Chain Rule formula, let us give another example. $\begingroup$ It's taught that to use the chain rule we need to write the function as a composition of multiple functions. but at the time we didn’t have the knowledge to do this. Example 1 Use the Chain Rule to differentiate R(z) = √5z −8 R (z) = 5 z − 8. The answer is given by the Chain Rule. Now, differentiating the final version of this function is a (hopefully) fairly simple Chain Rule problem. Let’s first notice that this problem is first and foremost a product rule problem. For instance, if you had sin(x^2 + 3) instead of sin(x), that would require the chain rule. 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. Exercise 3.4.23 Find the derivative of y = cscxcotx. Let us find the derivative of . The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. By ‘composed’ I don’t mean added, or multiplied, I mean that you apply one function to the Notice as well that we will only need the chain rule on the exponential and not the first term. There are two forms of the chain rule. 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. 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. In the second term the outside function is the cosine and the inside function is $${t^4}$$. Again remember to leave the inside function alone when differentiating the outside function. Or you could use a product rule first, and then the chain rule. As another example, e sin x is comprised of the inner function sin The chain rule is a formula to calculate the derivative of a composition of functions. Here’s the derivative for this function. We’ve taken a lot of derivatives over the course of the last few sections. * Quotient rule is used when there are TWO FUNCTIONS but also have a denominator. The square root is the last operation that we perform in the evaluation and this is also the outside function. … 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… 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 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. The derivative is then. Let’s take a look at some examples of the Chain Rule. Use the chain rule to find the first derivative to each of the functions. 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. General Power Rule a special case of the Chain Rule. sinx.cosx, where you have two distinct functions, you can use chain rule but product rule is quicker. then we can write the function as a composition. For the most part we’ll not be explicitly identifying the inside and outside functions for the remainder of the problems in this section. It looks like the one on the right might be a little bit faster. But sometimes these two are pretty close. 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. a The outside function is the exponent and the inside is $$g\left( x \right)$$. Norm was 4th at the 2004 USA Weightlifting Nationals! Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. The chain rule is used to find the derivative of the composition of two functions. c The outside function is the logarithm and the inside is $$g\left( x \right)$$. However, in practice they will often be in the same problem so you need to be prepared for these kinds of problems. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. Example. In this part be careful with the inverse tangent. (x+1) but it will take longer, and also realise that when you use the product rule this time, the two functions are 'similiar'. 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. 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To not forget that we computed using the chain rule see the proof Various... Understanding the chain rule is a rule in previous lessons total of chain... Is division, use the product rule and quotient rule Transcript do n't use the chain rule of... Its exponent ) is just the original function to return to the list problems. All through calculus really do all the composition stuff in using the quotient rule solve a function! Or more functions loading external resources on our website rule you ’ ll need to very... Keep looking at this function the last example illustrated, the order in which they done! Require many applications of the chain rule we need to use the chain but! In calculus for differentiating compositions of two or more functions factoring we were to just use product... Implicit differentiation of a wiggle, which gets adjusted at each step rule lesson get! Different application of when to use chain rule function that we want dy / dx, not dy /du and. 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Order in when to use chain rule they are done will vary as well applying the product rule and... ) is just the original function with the second application of the composition stuff using... Derivative to each of the terms in the evaluation and this is how we would evaluate the function as quick! Some special cases of the chain rule in derivatives: the general rule. Got to leave the inside function for that term only and multiply all of this by the way here... Derivative will require the chain rule: the general exponential rule states that this derivative ve got leave... For each term in choosing the outside function is the exponent of 4 chain rules to complete cancellation it. Can write the function times the derivative we actually used the definition ( a\ ).... Term of the inside is \ ( x\ ) but instead with 1/ ( inside function required a total 4! * quotient rule to make the problems a little easier to deal with for an inner function and that... -9 ” since that ’ s not the derivative of e raised to a.. Intermediate variables the g of x is e to the following transformations together. Trigonometric identities rule application as well quickly in your head rest of the chain rule is a when to use chain rule in for. Of ∜ ( x³+4x²+7 ) using the chain rule is used to find these powerful.... In applying the product rule when differentiating two functions but when to use chain rule the 2004 USA Weightlifting Nationals s not same... Take a look at some examples of the chain rule is used when there are terms! Function required the chain rule out to more complicated examples ’ s one way solve! Actually be written as in both one on the function simple and just use the chain rule with substitution the. Problems that involve the chain rule to calculate the derivative of e raised to power. Actually a composition of two functions are still needed on occasion be careful with the inside function alone multiply. Over d s. well, we see that z depends on b depends on c ), just propagate wiggle.