derivatives of products and quotients lesson 4.2
TRANSCRIPT
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Derivatives of Products and QuotientsLesson 4.2
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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.
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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
• Then3
( ) '( )xD 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 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
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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
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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 denominator 6
( )( )
( )
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
7
2 4
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
8
( )( )
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 items9
3 2( )
4
xC x
x
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Assignment
Lesson 4.2A
Page 259
Exercises 1 – 33 odd
Lesson 4.2B
Page 260
Exercises 39 – 49 odd
10