Section 2.6

Recall that the composition of two functions is performing the two function rules consecutively.  For example f(x) = (2x + 1)⁷ is the composition of the functions x⁷ and 2x + 1.   The function performed first, 2x + 1 in this example, is called the inside function.  The function performed second, x⁷ in this example, is called the outside function.

To differentiate a composition we use The Chain Rule, which consists of 3 steps:

1. Differentiate the outside function.

2. Don’t write x.  Where x goes, put the inside function.

3. Multiply by the derivative of the inside function.

Let’s follow these three steps to find the derivative of f(x) = (2x + 1)⁷.

1. The derivative of x⁷ is 7x⁶.

2. But we don’t write the x.   Where the x goes, we put 2x + 1.  This gives us 7(2x + 1)⁶.

3. We multiply by the derivative of 2x + 1, which is 2.  That makes our final answer 14(2x + 1)⁶.

If you missed class, you can watch my online movies. You must have QuickTime Player, a free download, installed on your computer.
Composition of functions is reviewed and the Chain Rule is introduced in movie 1 on this page.
Examples of using the Chain Rule can be found in movies 2 and 3 on this page.