the product rule complete the warm-up on your notesheet
TRANSCRIPT
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
What is the derivative of this function?
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
ConclusionConclusion
Remember: The derivative of a Remember: The derivative of a product is the first times the product is the first times the derivative of the second plus the derivative of the second plus the second times the derivative of the second times the derivative of the first.first.