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

We know:

But what about:

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.

Let Then:

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.

Find the derivative of the function

Suppose we know: Calculate