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
10
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