The Chain Rule

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 outside

The inside


The outside


The outside

The inside


The outside


Peel it away


There’s still more to peel.


Some functions have many layers that must be peeled away in order to find their derivatives.


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.


Outside function


Inside function


Outside function

2 3( )1xy


dydx

3 2( ) (2 )x2 1x


2 21 2(3( ) )dy x xdx

The derivative of the outside

The derivative of the inside (blop)


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


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


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


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


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.

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The Chain Rule