15 b the chain rule. we now have a small list of “shortcuts” to find derivatives of simple...

15BThe Chain Rule

We now have a small list of “shortcuts” to find derivatives of simple functions.

Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.

How would you go about finding the derivative of the following?

2( ) 3f x x2 23( ) 3x h x

h

2 2 23 6 3 3x xh h x

h

If h(x) = g(f(x)),

then h’(x) = g’(f(x))●f’(x).

The Chain Rule deals with the idea of composite functions and it is

helpful to think about an outside and an inside function when

using The Chain Rule.

2( ) 3f x x

23u x Inside Function

2( ) 3f x x

23u x Inside Function

y u Outside Function

In other words: The derivative when using the Chain Rule is the derivative of the outside leaving the inside unchanged times the derivative of the inside.

If h(x) = g(f(x)),

then h’(x) = g’(f(x))●f’(x).

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

dy dy du

dx du dx

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

1

21

62

dyu x

dx

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

123

dyxu

dx

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

12

3dy x

dx u

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

3dy x

dx u

Consider a simple composite function:

23y x

y u

2If 3u x

12y u 23u x

12

1

2

dyu

du

6

dux

dx

2

3

3

dy x

dx x

Find the derivative of 32 )3( xxf

Identify outside function and the inside function.

The outside function is the cube, ( )3

The inside function is x2 +3.

xxxf 233'22

The derivative of the inside using the Power Rule

The derivative of the outside leaving the inside unchanged

Next, simplify

22 36' xxxf

xxxf 233'22

Find the derivative2 71) ( ) (3 5 )f x x x

2 232) ( ) ( 1)f x x

2

73) ( )

(2 3)f t

t

Solutions2 71) ( ) (3 5 )f x x x

dy dy du

dx du dx

2

7 6

3 5 3 10

7

u x x x

y

du

dy

du

x

u u

d

6 ( 10 )7 3 xdy

udx

2 67(3 2 ) (3 10 )dy

x x xdx

Solutions2 232) ( ) ( 1)f x x

dy dy du

dx du dx

2

2 13 3

1 2

2

3

u x xd

dy u u

y

du

u

dx

132

3(2 )

du

y

dx

x

12 32

( 1)3

(2 )dy

dxx x

12 3

4

3( 1)

dy x

dxx

Solutions

dy dy du

dx du dx

2 3

2 3 2

7 14

u t

y u

d

d

d

u

d

u

x

uy

314 2u

dy

dx

33

2814(2 3) (2)

(2 3)

dyt

dx t

22

73) ( ) 7(2 3)

(2 3)f t t

t

Find the derivative of 3 2 13 xxh

To find the derivative of the outside, do the Power Rule:

3

2

13

1

3

1

3

13

1

withStarting

1

3 3Outside Function:

Find the derivative of 3 2 13 xxh

To find the derivative of the Inside, do the Power Rule:

2Inside Function: 3 1x

Inside Function: 2 3 6x x

xxxh 6133

1' 3

22

Now do a little simplification: Multiply the 1/3 and the 6x.

3 22

3

22

13

2or132'

x

xxxxh

Now let’s look at the actual derivative using the Chain Rule.

The derivative of the outside leaving the inside unchanged

The derivative of the inside

One Last Thought

It takes a big man to cry, but it takes a

bigger man to laugh at that man.

Homework

Page 364 (#1 – 2)

Page 366 (#1 – 6)