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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
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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
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( )( )
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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