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Quotients Rule!

4

This lesson will show us how to take

derivatives of products and quotients.

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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

Then5

( ) '( )

x

D k f x k f x

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

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

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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 thesecond plus the second times the derivative of

the first6

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

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

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Try It Out

Given the following functions which are

products

Determine the derivatives

7

25 1 4 3y x x

( ) 2 3 1f x x x

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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

denominator8

( )( )

( )

p xf x

q x

2

( ) '( ) ( ) '( )'( )

( )

q x p x p x q xf x

q x

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OK Try That!

Use the quotient rule on the following

functions

9

24

3

x xy

x

2.2

3.2( )

5

zh z

z

6 11

8 1

xy

x

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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

10

( )( )

C xC x

x

'( )C x

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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 items11

3 2( )4

xC xx

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Assignment

Lesson 4.2A

Page 259

Exercises 1 33 odd

Lesson 4.2B

Page 260

Exercises 39 49 odd

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