In this section, we will investigate how to take the derivative of a function that is the composition of multiple functions.

Section 3.2 The Chain Rule

The Idea

We know:

The Idea

We know:

But what about:

The Idea

These are all composition functions – that is multiple functions chained together. To take the derivative of such functions, we need to extend the “basic” rules with the chain rule.

TheoremThe Chain Rule

Let

Then:

TheoremThe Chain Rule

Let

Then:

What this really says:

We take the derivative of the “outside” function (leaving the “inside” function alone), and then multiply the result by the derivative of the “inside” function.

Example 1

Find the derivative of the function

Example 2

Find the derivative of the function

Example 3

Find the derivative of the function

Example 4

Find the derivative of the function

Example 5

Find the derivative of the function

Example 6

Find the derivative of the function

Example 7Numerical Example

Suppose we know:

Calculate