Lec5_ The Chain Rule.ppt

Objective
 To use the chain rule for
differentiation.
 ES: Explicitly assess information and
draw conclusions

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

The Chain Rule
( ( ))
y f g x

Inside function
Outside function
(derivative of outside function) (derivative of inside function)
dy
dx
 

The Chain Rule
If ( ( ))
y f g x

Then
dy
dx
 '( )
f ( )
g x '( )
g x


The Chain Rule
1. Identify inner and outer functions.
2. Derive outer function, leaving the
inner function alone.
3. Derive the inner function.

The Chain Rule (Example A)
2 3
( 1)
y x
 

The Chain Rule
2 3
( )
1
x
y  
Inside function
 Key Point: If the inside function contains
something other than plain old “x,” you must use
the Chain Rule to find the derivative.

The Chain Rule
2 3
( )
1
x
y  
Outside function

The Chain Rule
2 3
( )
1
x
y  
Inside function

The Chain Rule
Outside function
2 3
( )
1
x
y  

The Chain Rule
2 3
( 1)
y x
 
dy
dx
 3 2
( ) (2 )
x
2
1
x 

The Chain Rule
2 3
( 1)
y x
 
2 2
1 2
(
3( ) )
dy
x x
dx
 
The derivative
of the outside
The derivative
of the inside (blop)

The Chain Rule
2 3
( 1)
y x
 
2 2
3( 1) (2 )
dy
x x
dx
 
2 2
6 ( 1)
dy
x x
dx
 

The Chain Rule
1
3
4 2
Differentiate: ( ) (6 2 3)
f x x x
  
'( )
f x  1
3
2
3
( )
3
24 4
x x

CAUTION:
Possible mistakes ahead!
What mistake did I make?
I changed the inside function and did not multiply
by the derivative of the inside function.

Solve 1
3
4 2
f x x x
  
4 6
Differentiate: ( ) (3 2 )
f x x x
 
2
Differentiate: ( ) 3 1
g x x x
  
2
7
Differentiate: ( )
(2 3)
h x
x



2 60
f x x x
  
2 2
3
Differentiate: ( ) ( 1)
g x x
 
5 3
1
Differentiate: ( )
(2 7)
h x
x



The Chain Rule (Example B)
1
3
4 2
f x x x
  
'( )
f x  1
3
2
3
( )
4 2
6 2 3
x x
  3
(24 4 )
x x

'( )
f x  1
3
2
3
( )
4 2
6 2 3
x x
  2
(4 )(6 1)
x x 
'( )
f x  4
3 x
2
3
( )
4 2
6 2 3
x x
  2
(6 1)
x 
 
 
 
2
2
4 2 3
4 6 1
'
3 6 2 3
x x
f x
x x


 

The Chain Rule (Example C)
4 6
Differentiate: ( ) (3 2 )
f x x x
 
'( )
f x  6
5
( )
4
3 2
x x
 3
(3 8 )
x


The Chain Rule (Example D)
2
Differentiate: ( ) 3 1
g x x x
  
'( )
g x  1
2
1
2
( )

2
3 1
x x
  (6 1)
x 
1
2 2
( ) (3 1)
g x x x
  
'( )
g x 
2
1
2 2
(3 1)
x x
 
6 1
x 
2
6 1
'( )
2 3 1
x
g x
x x


 

The Chain Rule (Example E)
2
7
Differentiate: ( )
(2 3)
h x
x



'( )
h x  14 3
( )
2 3
x  (2)
2
( ) 7(2 3)
h x x 
  
'( )
h x  3
(2 3)
x 
28

The Chain Rule (Example F)
2 60
f x x x
  
'( )
f x  60
59
( )
2
2 4 1
x x
  (4 4)
x 
'( )
f x  60
59
( )
2
2 4 1
x x
  (4)( 1)
x 
'( )
f x  240
59
( )
2
2 4 1
x x
  ( 1)
x 

The Chain Rule (Example G)
2 2
3
Differentiate: ( ) ( 1)
g x x
 
'( )
g x  2
3
1
3
( )

2
1
x  (2 )
x
2
2 3
( ) ( 1)
g x x
 
'( )
g x 
3
1
2 3
( 1)
x 
2
3 2
4
'( )
3 1
x
g x
x


(2 )
x

The Chain Rule (Example H)
5 3
1
Differentiate: ( )
(2 7)
h x
x


'( )
h x  3

4
( )
5
2 7
x  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)
x
h x
x




The Chain Rule (Example I)
2
3
Differentiate: ( )
1
j x
x


'( )
j x  3
 2
( )
2
1
x  (2 )
x
2 1
( ) 3( 1)
j x x 
 
'( )
j x  2 2
( 1)
x 
3
 (2 )
x
2 2
6
'( )
( 1)
x
j x
x




Conclusion
 Remember: When a function is inside
another function, use the Chain Rule to
find the derivative.
 First, differentiate the outside function,
leaving the inside function alone.
 Last, differentiate the inside function.

Lec5_ The Chain Rule.ppt

More Related Content

Similar to Lec5_ The Chain Rule.ppt (20)

Recently uploaded (20)

Lec5_ The Chain Rule.ppt