the chain rule
DESCRIPTION
The Chain Rule. Mr. Miehl [email protected]. Objective. To use the chain rule for differentiation. The Chain Rule. The Chain Rule is a technique for differentiating composite functions . Composite functions are made up of layers of functions inside of functions. The Chain Rule. - PowerPoint PPT PresentationTRANSCRIPT
The Chain RuleThe Chain RuleMr. MiehlMr. Miehl
[email protected]@tesd.net
ObjectiveObjective To use the chain rule for To use the chain rule for
differentiation.differentiation.
The Chain RuleThe Chain Rule The Chain Rule is a technique for The Chain Rule is a technique for
differentiating differentiating composite functionscomposite functions..
Composite functionsComposite functions are made up of are made up of layers of functions inside of layers of functions inside of functions.functions.
The Chain RuleThe Chain Rule
The outside
The Chain RuleThe Chain Rule
The outside
The inside
The Chain RuleThe Chain Rule
The outside
The Chain RuleThe Chain Rule
The outside
The inside
The Chain RuleThe Chain Rule
The outside
The Chain RuleThe Chain Rule
Peel it away
The Chain RuleThe Chain Rule
There’s still more to peel.
The Chain RuleThe Chain Rule
Some functions have many layers that must be peeled away in order to find their derivatives.
The Chain RuleThe Chain Rule
The Chain RuleThe Chain Rule( ( ))y f g x
Inside function
Outside function
(derivative of outside function) (derivative of inside function)dydx
The Chain RuleThe Chain Rule2 3( 1)y x
The Chain RuleThe Chain Rule2 3( )1xy
Inside function
Key Point: If the inside function contains Key Point: If the inside function contains something other than plain old “something other than plain old “xx,” you must use ,” you must use the Chain Rule to find the derivative.the Chain Rule to find the derivative.
The Chain RuleThe Chain Rule2 3( )1xy
Outside function
The Chain RuleThe Chain Rule2 3( )1xy
Inside function
The Chain RuleThe Chain Rule
Outside function
2 3( )1xy
The Chain RuleThe Chain Rule2 3( 1)y x
dydx
3 2( ) (2 )x2 1x
The Chain RuleThe Chain Rule2 3( 1)y x
2 21 2(3( ) )dy x xdx
The derivative of the outside
The derivative of the inside (blop)
The Chain RuleThe Chain Rule2 3( 1)y x
2 23( 1) (2 )dy x xdx
2 26 ( 1) dy x xdx
The Chain RuleThe Chain RuleIf ( ( ))y f g x
Then dydx
'( )f ( )g x '( )g x
The Chain RuleThe Chain Rule
1. Identify inner and outer functions.1. Identify inner and outer functions.2. Derive outer function, leaving the 2. Derive outer function, leaving the
inner function alone.inner function alone.3. Derive the inner function.3. Derive the inner function.
The Chain RuleThe Chain Rule1
34 2Differentiate: ( ) (6 2 3)f x x x
'( )f x 13
23( )324 4x x
CAUTION: Possible mistakes ahead!What mistake did I make?
I changed the inside function and did not multiple by the derivative of the inside function.
The Chain RuleThe Chain Rule1
34 2Differentiate: ( ) (6 2 3)f x x x
'( )f x 13
23( )4 26 2 3x x 3(24 4 )x x
'( )f x 13
23( )4 26 2 3x x 2(4 )(6 1)x x
'( )f x
234 26 2 3 x x3
2(4 )(6 1)x x
The Chain RuleThe Chain Rule4 6Differentiate: ( ) (3 2 )f x x x
'( )f x 6 5( )43 2x x 3(3 8 )x
The Chain RuleThe Chain Rule2Differentiate: ( ) 3 1g x x x
'( )g x 12
12( )23 1x x (6 1)x
12 2( ) (3 1)g x x x
'( )g x 2
12 2(3 1)x x
6 1x
2
6 1'( )2 3 1
xg xx x
The Chain RuleThe Chain Rule
2
7Differentiate: ( )(2 3)
h xx
'( )h x 14 3( )2 3x (2)
2( ) 7(2 3)h x x
'( )h x 3(2 3)x
28
The Chain RuleThe Chain Rule2 60Differentiate: ( ) (2 4 1)f x x x
'( )f x 60 59( )22 4 1x x (4 4)x
'( )f x 60 59( )22 4 1x x (4)( 1)x
'( )f x 240 59( )22 4 1x x ( 1)x
The Chain RuleThe Chain Rule2 23Differentiate: ( ) ( 1)g x x
'( )g x 23
13( )2 1x (2 )x
22 3( ) ( 1)g x x
'( )g x 3
12 3( 1)x
2
3 2
4'( )3 1
xg xx
(2 )x
The Chain RuleThe Chain Rule
5 3
1Differentiate: ( )(2 7)
h xx
'( )h x 3 4( )52 7x 4(10 )x
5 3( ) (2 7)h x x
'( )h x 5 4(2 7)x
3 4(10 )x
4
5 4
30'( )(2 7)
xh xx
The Chain RuleThe Chain Rule
2
3Differentiate: ( )1
j xx
'( )j x 3 2( )2 1x (2 )x
2 1( ) 3( 1)j x x
'( )j x 2 2( 1)x
3 (2 )x
2 2
6'( )( 1)
xj xx
ConclusionConclusion Remember: When a function is inside Remember: When a function is inside
another function, use the Chain Rule to another function, use the Chain Rule to find the derivative.find the derivative.
First, differentiate the outside function, First, differentiate the outside function, leaving the inside function alone. leaving the inside function alone.
Last, differentiate the inside function.Last, differentiate the inside function.