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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