in this section, we will investigate how to take the derivative of a function that is the...
TRANSCRIPT
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