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The Product RuleThe Product Rule
Complete the warm-up Complete the warm-up on your notesheeton your notesheet
ObjectiveObjective
To use the product rule for To use the product rule for differentiation.differentiation.
TS: Making decisions after reflection TS: Making decisions after reflection and reviewand review
The Product RuleThe Product Rule
35y xA number multiplied by A number multiplied by a variable raised to a power.a variable raised to a power.
Bad News:Bad News: The Power Rule The Power Rule doesn’t work for a lot of derivatives.doesn’t work for a lot of derivatives.
Good News:Good News: Right now you can find the Right now you can find the derivative of functions like …derivative of functions like …
The Product RuleThe Product Rule2(3 2 ) (5 4 )y x x x
This function consists of This function consists of two separate piecestwo separate piecesmultiplied togethermultiplied together
The Product RuleThe Product Rule2(3 2 ) (5 4 )y x x x
dy
dxDoesDoes (3 4 ) x (4) ?? NO!
Take each derivative
The Product RuleThe Product Rule
The derivative of a product The derivative of a product is not equal to the product is not equal to the product
of the derivatives.of the derivatives.
The Product RuleThe Product Rule
The derivative of a product must by The derivative of a product must by calculated using the product rule:calculated using the product rule:
then ( ) '( ) ( ) '( )dy
f x g x g x f xdx
If ( ) ( )y f x g x
The Product RuleThe Product Rule
1.1. Imagine that the function is actually Imagine that the function is actually broken into 2 pieces.broken into 2 pieces.
2(3 2 ) (5 4 )y x x x
2(3 2 )x x (4)
The Product RuleThe Product Rule
2. Leave first piece alone and derive 2. Leave first piece alone and derive second piece.second piece.
dy
dx
2(3 2 ) (5 4 )y x x x
The Product RuleThe Product Rule
3. Add: Leave second piece alone and 3. Add: Leave second piece alone and derive first piece.derive first piece.
dy
dx (3 4 )x(5 4 )x2(3 2 )x x (4)
2(3 2 ) (5 4 )y x x x
The Product RuleThe Product Rule
dy
dx 2(12 8 )x x (15 20x 12x 216 )x
dy
dx 224x 4x 15
This is the derivative!
dy
dx (3 4 )x(5 4 )x2(3 2 )x x (4)
The Product Rule Example A:The Product Rule Example A:1( 1)( 1)xy x
1( 1)( 1)y x x
dy
dx 1( 1)x (1) ( 1)x 2( )x
The Product RuleThe Product Rule
dy
dx 1( 1)x (1) ( 1)x 2( )x
dy
dx
1
x1
2
1x
x
dy
dx 2x 2x
2
1x
x
x 2x
2
2
1dy x
dx x
The Product Rule Examle BThe Product Rule Examle B23( ) ( 4)h x x x
1 23( ) ( 4)h x x x
'( )h x 1
3x (2 )x 2( 4)x 2
1 33( )x
The Product RuleThe Product Rule
'( )h x 1
3x (2 )x 2( 4)x 2
1 33( )x
'( )h x 4
32x
2 4x 3
23x
2 23 3
'( )
3 3h x
x x
2 4x 26x
2
23
7 4'( )
3
xh x
x
The Product Rule Example CThe Product Rule Example C3 2( ) (1 4 )(3 5 2)f x x x x
'( )f x 3(1 4 )x (6 5)x 2(3 5 2)x x 2( 12 )x
'( )f x 6x 5 424x 320x 436x 360x 224x
'( )f x 460x 380x 224x 6x 5