Module 6 the chain rule
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The chain rule provides a formula for finding the derivative of a composite function h(x) = f(g(x)). The formula is: h'(x) = f'(g(x))g'(x). This takes the derivative of the outer function f with respect to the inner function g, and multiplies it by the derivative of the inner function g. An example applies the chain rule to find the derivative of h(x) = (x3 + 4x)3/2 by identifying the inner and outer functions, taking their individual derivatives, and plugging into the chain rule formula.
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![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]](https://image.slidesharecdn.com/module6thechainrule-210225010723/85/Module-6-the-chain-rule-1-320.jpg)
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]