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MAT 1221Survey of Calculus
Section 2.4
The Product and Quotient Rules
http://myhome.spu.edu/lauw
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Expectations
Use equal signs. Show necessary steps. Double check your calculator
calculations.
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Recall: Sum and Difference Rule
If , then)()( xvxuy )()( xvxuy
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However,…
If , then
If , then
)()( xgxfy ( ) ( )y f x g x
)(
)(
xg
xfy
)(
)(
xg
xfy
?Why
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Example 1(a)2( ) ( ) where ( ) 4, ( )y f x g x f x x g x x
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Example 1(b)
2( ) where ( ) 4, ( )
( )
f xy f x x g x x
g x
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However,…
If , then
If , then
)()( xgxfy ( ) ( )y f x g x
)(
)(
xg
xfy
)(
)(
xg
xfy
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Correct Formulas???
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Product Rule
If ,
then
)()( xgxfy
( ) ( ) ( ) ( )y f x g x f x g x
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Example 2 (1(a)) Verify the Product Rule
2( ) ( ) where ( ) 4, ( )y f x g x f x x g x x
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Expectation
Show the “formula” step
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Example 3
2(4 1)(3 1)y x x
dy
dx
( ) ( )
( ) ( ) ( ) ( )
y f x g x
y f x g x f x g x
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Remark
Should we use the product rule?
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Quotient Rule
If
then
)(
)(
xg
xfy
2
( ) ( ) ( ) ( )
( )
f x g x f x g xy
g x
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Example 4 (1(b))Verify the Quotient Rule
2( ) where ( ) 4, ( )
( )
f xy f x x g x x
g x
2)(
)()()()(
xg
xgxfxgxfy
2 2
2
4 4x x x xy
x
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Expectation
2 2
2
4 4x x x xy
x
Show the “formula” step
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Example 52
2 1
xy
x
y
2)(
)()()()(
xg
xgxfxgxfy
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Remarks
Sometimes, it is easier to NOT use the quotient rule
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Example 7
2
7yt
dy
dt
2
7yt
dy
dt
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Expectations
Must show the “formula” step
Double check your algebra!!! Read the hints Use equal signs
2 2
2 2
2
( ) (4 1) (3 1) (4 1)(3 1)
2 1 2 1
2 1
f x x x x x
x x x xy
x