Derivatives of Products and QuotientsLesson 4.2

Quotients Rule!

2

Products Rule!

This lesson will show us how to take derivatives of products and quotients.

This lesson will show us how to take derivatives of products and quotients.

Review

We know what to do with constant times a function• For k • f(x)

We also know what to do with the sum of functions• When

• Then3

( ) '( )xD k f x k f x

( ) ( ) ( )f x h x k x

'( ) '( ) '( )f x h x k x

Product Rule

Consider the product of two functions

It can be shown (see proof, pg 215) that

In words:• The first function times the derivative of the

second plus the second times the derivative of the first 4

( ) ( ) ( )f x h x k x

'( ) ( ) '( ) ( ) '( )f x h x k x k x h x

Try It Out

Given the following functions which are products• Determine the derivatives

5

25 1 4 3y x x

( ) 2 3 1f x x x

Quotient Rule

When our function is the quotient of two other functions …

The quotient rule specifies the derivative

In words:• The denominator times the derivative of the numerator

minus the numerator times the derivative of the denominator, all divided by the square of the denominator 6

( )( )

( )

p xf x

q x

2( ) '( ) ( ) '( )

'( )( )

q x p x p x q xf x

q x

OK … Try That!

Use the quotient rule on the following functions

7

2 4

3

x xy

x

2.2

3.2( )

5

zh z

z

6 11

8 1

xy

x

Average Cost

Suppose we have a function y = C(x) which gives us the cost of manufacturing x items

The average cost is

Then the marginal average cost is

8

( )( )

C xC x

x

'( )C x

Average Cost

Suppose the cost for manufacturing x items is

Write the function for the average cost

What is the marginal average cost?• Determine rate of change of the average cost

for 5 items … for 50 items9

3 2( )

4

xC x

x

Assignment

Lesson 4.2A

Page 259

Exercises 1 – 33 odd

Lesson 4.2B

Page 260

Exercises 39 – 49 odd

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