1. 11.4 The Chain Rule

2. Composite Functions
Definition: A function m is a composite of functions f and g if
m(x) = f [g(x)]
The domain of m is the set of all numbers x such that x is in the domain of g and g(x) is in the domain of f.

3. Example 1 Given f(u) = 2u and g(x) = ex Find f [g(x)] and g [f(u)]
f [g(x)] = f (ex) substitute g(x) for ex
= 2(ex) plug ex into function f(u)
= 2ex
g [f(u)] = g (2u) substitute f(u) for 2u
= e2u plug 2u into function g(x)

4. Example 2 Write each function as a composite of two simpler functions
A) Y = 50 e-2x
g(x) = -2x and f(u) = 50 eu B)
and

5. C) Write y as a composite of two simpler functions
y = (2x + 1)100
g(x) = 2x and f(u) = u 100
How do we find the derivative of a composite
function such as the one above?
What is the derivative of (2x + 1)100
Let’s investigate on the next slide!

6. Find the derivative of (2x + 1)2, (2x + 1)3, (2x + 1)4 and (2x + 1)100
Use the product rule:
F(x) = (2x + 1)2 = (2x + 1) (2x + 1)
F’(x) = (2)(2x+1) + (2)(2x+1) = 4(2x+1)
F(x) = (2x + 1)3 = (2x + 1) (2x + 1)2 = (2x+1)(4x2 + 4x +1)
F’(x) = (2)(4x2 + 4x +1) + (8x + 4)(2x + 1)
= (2x+1) (2x+1)(2x+1)
= (2x+1) (2x+1)2 = 6 (2x+1)2
Try (2x + 1)4 you should find out that f’(x) = 8 (2x+1)3
Look at the relationship between f(x) and f’(x), do you agree that the derivative of (2x + 1)100 is 200 (2x+1)99 ?

7. f(x) f’(x) (2x + 1)2 4(2x+1) = 2(2x+1) · 2 (2x + 1)3
Any composite function [g(x)] n
n·[g(x)]n-1 · g’(x)

8. Generalized Power Rule
What we have seen is an example of the generalized power rule: If u is a function of x, then

9. Chain Rule
We have used the generalized power rule to find derivatives of composite functions of the form f (g(x)) where f (u) = un is a power function. But what if f is not a power function? It is a more general rule, the chain rule, that enables us to compute the derivatives of many composite functions of the form f(g(x)).
Chain Rule: If y = f (u) and u = g(x) define the composite function y = m(x) = f [g(x)], then
Or m’(x) = f’[g(x)] g’(x) provided that f’[g(x) and g’(x) exist

10. Example 3 A) h’(x) if h(x) = (5x + 2)3 h’(x) = 3 (5x + 2)2 (5x+2)’
Find the indicated derivatives
A) h’(x) if h(x) = (5x + 2)3
h’(x) = 3 (5x + 2)2 (5x+2)’
= 3 (5x + 2)2 (5) =15 (5x + 2)2
B) y’ if y = (x4 – 5)5
y’= 5(x4 – 5)4 (x4 – 5)’
= 5(x4 – 5)4 (4x3) = 20x3 (x4 – 5)4

11. Example 3
Find the indicated derivatives C) D)

12. Example 4 A) y = u-5 and u = 2x3 + 4
Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)
A) y = u-5 and u = 2x3 + 4
The problem above can be solved by taking the derivative of y = (2x3 + 4) -5

13. Example 4 (continue) B) y = eu and u = 3x4 + 6
Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)
B) y = eu and u = 3x4 + 6
This problem can be solved by taking the derivative of

14. Example 5 C) y = ln u and u = x2 + 9x + 4
Find dy/du, du/dx, and dy/dx (express dy/dx as a function of x)
C) y = ln u and u = x2 + 9x + 4
This problem can be solved by taking the derivative of y = ln (x2 + 9x + 4)

15. The chain rule can be extended to compositions
of three or more functions.
Given: Y = f(w), w = g(u), and u = h(x)
Then

16. Example 6 y = h(x) = [ ln(1 + ex) ]3 find dy/dx
Let y = w 3, w = ln u, and u = 1 + ex
then

17. Example 6 (continue) y = h(x) = [ ln(1 + ex) ]3 find dy/dx Another way
Derivative of the outside function
The derivative of the function inside the [ ]
Derivative of the function inside ( )

18. Examples for the Power Rule
Chain rule terms are marked:
Compare with the next slide

19. Examples for the Power Rule
Chain rule terms are marked:

20. Examples for Exponential Derivatives
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21. Examples for Exponential Derivatives
Property of ln

22. Examples for Logarithmic Derivatives
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23. Examples for Logarithmic Derivatives