2011 – Chain Rule

2011 – Chain Rule

AP CALCULUS

If you can’t do Algebra . . . .

I guarantee yousomeone will do Algebra on you!

COMPOSITE FUNCTIONS

Know: Need:

Know: Need:  REM:

2( 2)y x 8( 2)y x 2( 2)y x 4 3 2 2(4 3 2 7 5)y x x x x

2010

10

f(x)=1.0825(x)

g(x)=.5(x)

COMPOSITE FUNCTIONS

( ( ))y f g x Let and

y = outside u = inside

function function

( )u g x( )y f u

g(x) - what is done first.

f(x)

x

COMPOSITE FUNCTIONS

DECOMPOSE y = (outside) u = (inside).

2 3( 1)y x

23 1y x x

1

1y

x

tan 2y x2tany x

2tany x

( ( ( ))d

f g xdx

In Words:

__________________________________________________________

 Derivative of __________________________________________________

Chain Rule

( )u g x

Leibniz Notation:

In Words:

__________________________________________________________

( ( ))y f g x ( )y f u

Example 1:

Example:

Let y = u =

  _______

dy

du

du

dx

dy

dx

2 3(3 2 1)y x x

Example 2:2 3(3 2 )y x x

Example 3:

23 ( 2)y x

Example 4:

Example:1

(2 3)y

t

2

7

(3 5)y

x

Note: 2

1

4(3 5)y

x

Example 6:

cos(3 )y x

2cosy x

Example 7:

2 3sin ( )y x

Extended Chain:

Ex: OR

WORDS:

Extended Chain: ___________

Derivative of the Absolute Value Function

REM:2x x

2d dx x

dx dx Do not simplify.

Use the Chain Rule.

General Rules: Working with number values

Find the derivative.

1) f(x) + g(x) at x = 3 2) 2f(x) – 3g(x) at x =2

3) f(x)*g(x) at x = 2 4) f(x) / g(x) at x = 3

5) f(g(x)) at x = 2 6) (f(x))3 at x = 3

x f(x) g(x) f / (x) g / (x)

2 8 3 -4

3 2 -6 -512

23

Last Update

• 10/13/07

• Assignment p. 153 # 13 – 31 odd, 56