The Chain (Saw) Rule

Lesson 3.4

The Chain Rule• According to Mrs. Armstrong …

“Pull the chain and the light comes on!”

Introduction

Sludge Falls

• CO2 is changing at rate of 0.02 ppm for each person

• Population growing atrate of 1000 people/yr

• We seek rate of increasing pollution with respect to time

(0.02 ppm/prsn)(1000 people/yr) = 20 ppm/yr

A Composite Function

The level of pollution L is a function of the population P, which is itself a function of time t.

L = f(P(t))

Then L’ … is

Rate of Change of L with

respect to t=

Rate of change of L with

respect to P

Rate of change of P with

respect to t

A Composite FunctionIn Leibniz notation:

dL dL dP

dt dP dt

Result in pollution as a

function of time

Result in pollution as a

function of time Pollution as a function of the

population

Pollution as a function of the

population

Population as a function of

time

Population as a function of

time

The Chain RuleGiven • y = f(u) is a differentiable function of u• u is also a differentiable function … of x• Then y = f(u(x))

• Then

dy dy du

dx du dx

Example• Given:

y = (6x3 – 4x + 7)3

• Then u(x) = 6x3 – 4x + 7and f(u) = u3

• Thusf’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)

dy

du

du

dx

Example• Given g(u) = u5

u(x) = 3x + 1

• Then g’(u) = ?? u’(x) = ??

• f(x) = (3x + 1)5

f’(x) = ??

Example• Find equation of tangent line to

at (2,3)

2( ) 5f x x

Example

• Try

• Which is the u(x), the “inner” function?

• Which is the f(u), the “outer” function?

• What is u’(x), f’(u) ??

2cos(4 7 5)d

x xdx

Example• Try with multiple levels of nested functions

2 3( ) sin ((4 2) )f x x

Example• Try in combination with the product rule

2 3( ) (6 2) xg x x e

Assignment Part A

• Lesson 3.4A

• Page 161

• Exercises 1 – 49 Odd

Derivative for ln x

• Graph the difference function for ln x

This has a familiar look,

pardner!

This has a familiar look,

pardner!

1ln for x 0

dx

dx x

1ln for x 0

dx

dx x

Derivatives for Bases Other Than e

ln

ln

1log

ln

1log

ln

x x

u u

a

a

da a a

dxd dua a a

dx dxd

xdx a x

d duu

dx a u dx

ViewGeogebra

Demo

ViewGeogebra

Demo

ViewGeogebra

Demo

ViewGeogebra

Demo

One More Try

• Determine the derivative

• What is the equation of the tangent line of f(x) when x = 1?

ln( ) '( ) ?

tf t f t

t

( ) lnf x x x

Assignment Part B

• Lesson 3.4B

• Page 162

• Exercises 51 – 127 EOO