Section 12.2Derivatives of Products and Quotients

Product Rule

( ) ( ) '( ) ( ) '( ) ( )d

f x g x f x g x g x f xdx

Alternative Form: ' 'd

u v u v v udx

Examples

4 2( 3)y x x 2( 2)( 3 4)y x x x

3 2(5 4 30)y t t

Find y’ for the function.

Quotient Rule

2

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

( ) [ ( )]

d f x f x g x g x f x

dx g x g x

2

' 'd u u v v u

dx v v

Alternative Form:

Examples

Find dy/dx for the function.29 6

3

x xy

x

47 2 5

3 1

x xy

x

(3 5)( 1)x xy

x

Average Cost

If C(x) is a cost function, then the average cost function, is the function

The derivative of the average cost function is called the marginal average cost function.

( )( )

C xC x

x

Average Revenue / Profit If R(x) is a revenue function, then the

average revenue function is

If P(x) is a profit function, then the average profit function is

Recall Marginal function = derivative of that function

( )( )

R xR x

x

( )( )

P xP x

x

ExampleThe total profit (in tens of dollars) from selling x self-help books is:

Find the average profit from each sales level.

a) 8 books

b) 15 books

c) x books

d) Find the marginal average profit function

32

65)(

x

xxP