Module 6 the chain rule

1. THE CHAIN RULE
If 𝒇 and 𝒈 are differentiable functions, then the composite function 𝒉(𝒙) = 𝒇(𝒈(𝒙)) has derivative
given by:
𝒉′(𝒙) = 𝒇′(𝒈(𝒙))𝒈′(𝒙)
EXAMPLE 1: Differentiate 𝒉(𝒙) = (𝒙𝟑
+ 𝟒𝒙)
𝟑
𝟐
SOLUTION:
STEP 1 Identify the inner function and outer function
The inner function is: 𝒈(𝒙) = 𝒙𝟑
+ 𝟒𝒙
The outer function is: 𝒇(𝒙) = 𝒙
𝟑
𝟐
STEP 2 Solve for the derivative of the inner function and the outer function
• The inner function
𝒈(𝒙) = 𝒙𝟑
+ 𝟒𝒙
𝒈′(𝒙) = 𝟑𝒙𝟐
+ 𝟒
• The outer function
𝒇(𝒙) = 𝒙
𝟑
𝟐
𝒇′(𝒙) =
𝟑
𝟐
𝒙
𝟏
𝟐
STEP 3 Use the Chain Rule Formula
𝒉′(𝒙) = 𝒇′(𝒈(𝒙))𝒈′(𝒙)
𝒉′(𝒙) = (
𝟑
𝟐
𝒙
𝟏
𝟐)( 𝟑𝒙𝟐
+ 𝟒)
𝒉′(𝒙) = [
𝟑
𝟐
(𝒙𝟑
+ 𝟒𝒙)
𝟏
𝟐][𝟑𝒙𝟐
+ 𝟒]