The Chain Rule

Suggested Prerequesites: The definition of the derivative

Suppose we want to find the derivative of y=(x2+3x+1)2. We could hopefully multiply it out and then take the derivative with little difficulty. But, what if, instead, it was y=(x2+3x+1)50 ? Would you want to apply the same method to this problem? Certainly not. Instead we need a method for dealing with composite functions, functions which are one function applied to another. For example, if we let u=f(x)=(x2+3x+1) and g(u)=u2 then y=(x2+3x+1)2 = g(f(x)). This is sometimes written as . This is read "g composite f."

Our goal is to find the derivative based on our knowledge of the functions f and g. Now, we know that

and

Leibniz's differential notation suggests that perhaps derivatives can be treated as fractions, leading to the speculation that

This leads to the (possible) chain rule:

Let's apply this to our example and see if it works. First, we'll multiply the product out and then take the derivative. Then we'll apply the chain rule and see if the results match:

So, our rule checks out, at least for this example. It turns out that this rule holds for all composite functions, and is invaluable for taking derivatives.

This rule is called the chain rule because we use it to take derivatives of composties of functions by chaining together their derivatives. The chain rule can be though 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 discussion given here is not by any means a proof and should not satisfy any reader. A proof of the chain rule can be found here. Please look at it.

Failure to apply the chain rule is probably the most common mistake in differential calculus. Remember that the chain rule applies to all composite functions.

Some examples:

  1. sin(2x) is the composite of the functions sin(x) and 2x. Then

    To check:

  3. The chain rule can be extended to composites of more than two functions. For example sin2(4x) is a composite of three functions -- sin(x), x2 and 4x. Just remeber, the derivative of the outer times the derivative of the inner times ...

Exercises:

    Take the derivatives of the following functions:

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page
jjnichol@mit.edu
Last Modified 23 June 1997