2011 – Chain Rule
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Transcript of 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