the chain rule section 2.4 after this lesson, you should be able to: find the derivative of a...
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The Chain Rule
Section 2.4
After this lesson, you should be able to:
• Find the derivative of a composite function using the Chain Rule.
• Find the derivative of a function using the General Power Rule.
• Simplify the derivative using algebra.• Find the derivative of a trigonometric function
using the Chain Rule.
The Chain Rule(deals with composition of functions)
If f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and
dy dy du
dx du dx
or,
[ ( ( ))] '( ( )) '( )df g x f g x g x
dx
Extended Power Rule
(power rule)
1)( nn
nxdx
xd 1)( nn
nudu
ud
(extended (or general) power rule)
'1unuudx
d nn
Example• Given:
y = (6x3 – 4x + 7)3
• Then u(x) = 6x3 – 4x + 7and f(u) = u3
• Thusf’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)
dy
du
du
dx
• Try
• Which is the, the “inner” function?
• Which is the , the “outer” function?
• What is the answer ??
2cos(4 7 5)d
x xdx
• Try
• Which is the, the “inner” function?=
• Which is the , the “outer” function?
• What is the answer ??
2cos(4 7 5)d
x xdx
24 7 5x x
• Try
• Which is the, the “inner” function? u =
• Which is the , the “outer” function? =
• What is the answer ??
24 7 5x x
2cos(4 7 5)d
x xdx
cos( )u
2 ' 2cos(4 7 5) cos( ) sin(4 7 5) (8 7)d d
x x u u x x xdx du
Example
Example:2802 )543( xxy
Example
Example:
2 280 2 279(3 4 5) 280(3 4 5) (6 4)y x x x x x
Example
Example: 34)( 2 xxxf
Example
Example:
12 ' 2 2
1( ) 4 3 ( ) ( 4 3) (2 4)
2f x x x f x x x x
Example
Example:1
2 ' 2 2
2
1( ) 4 3 ( ) ( 4 3) (2 4)
24
4 3
f x x x f x x x x
x
x x
Example
Example:10463 )22()3( xxxy
Example
Example:3 6 4 10
' 3 5 2 4 10 4 9 3 3 6
( 3) (2 2)
6( 3) (3 )(2 2) 10(2 2) (8 1)( 3)
y x x x
y x x x x x x x x
Example
Example:
3 6 4 10
' 3 5 2 4 10 4 9 3 3 6
2 3 5 4 10 4 9 3 3 6
( 3) (2 2)
6( 3) (3 )(2 2) 10(2 2) (8 1)( 3)
18 ( 3) (2 2) 10(2 2) (8 1)( 3)
y x x x
y x x x x x x x x
x x x x x x x x
Example
Example:3 6 4 10
' 3 5 2 4 10 4 9 3 3 6
2 3 5 4 10 4 9 3 3 6
3 5 4 9 2 4 3 3
( 3) (2 2)
6( 3) (3 )(2 2) 10(2 2) (8 1)( 3)
18 ( 3) (2 2) 10(2 2) (8 1)( 3)
2( 3) (2 2) 9 (2 2) 5(8 1)( 3)
y x x x
y x x x x x x x x
x x x x x x x x
x x x x x x x x
Example
Example:3 6 4 10
' 3 5 2 4 10 4 9 3 3 6
2 3 5 4 10 4 9 3 3 6
3 5 4 9 2 4 3 3
3 5 4 9 6
( 3) (2 2)
6( 3) (3 )(2 2) 10(2 2) (8 1)( 3)
18 ( 3) (2 2) 10(2 2) (8 1)( 3)
2( 3) (2 2) 9 (2 2) 5(8 1)( 3)
2( 3) (2 2) 18 9
y x x x
y x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x
2 6 318) 40 125 15x x x
Example
Example:3 6 4 10
' 3 5 2 4 10 4 9 3 3 6
2 3 5 4 10 4 9 3 3 6
3 5 4 9 2 4 3 3
3 5 4 9 6
( 3) (2 2)
6( 3) (3 )(2 2) 10(2 2) (8 1)( 3)
18 ( 3) (2 2) 10(2 2) (8 1)( 3)
2( 3) (2 2) 9 (2 2) 5(8 1)( 3)
2( 3) (2 2) 18 9
y x x x
y x x x x x x x x
x x x x x x x x
x x x x x x x x
x x x x
2 6 3
3 5 4 9 6 3 2
18 40 125 15
2( 3) (2 2) 58 125 9 3
x x x
x x x x x x
Example
Example:5
2
sin( )
1
x xf x
x
Example
Example:5
2
4 2'
2 2 2
sin( )
1
sin (sin cos )( 1) (2 ) sin( ) 5
1 ( 1)
x xf x
x
x x x x x x x x xf x
x x
Example
Example:5
2
4 2'
2 2 2
4 2 3 2
2 2 2
sin( )
1
sin (sin cos )( 1) (2 ) sin( ) 5
1 ( 1)
sin ( sin sin cos cos ) (2 sin )5
1 ( 1)
x xf x
x
x x x x x x x x xf x
x x
x x x x x x x x x x x
x x
Example
Example:5
2
4 2'
2 2 2
4 2 3 2
2 2 2
4 2 3
2
sin( )
1
sin (sin cos )( 1) (2 ) sin( ) 5
1 ( 1)
sin ( sin sin cos cos ) (2 sin )5
1 ( 1)
sin ( sin sin cos cos )5
1 (
x xf x
x
x x x x x x x x xf x
x x
x x x x x x x x x x x
x x
x x x x x x x x x
x x
2 2, ...
1)etc
Example
Example:xy 3sin
Example
Example:
3 ' 2sin 3sin ( ) cos( )y x y x x
Example
dx
duu
dx
udcos
sin
Example: )4sin(3 xy
Example
dx
duu
dx
udcos
sin
Example:
'3sin( 4) 3 cos( 4)y x y x
Example
dx
duu
dx
ud sin
cos
xxxf 3cos)( Example:
Example
dx
duu
dx
ud sin
cos
'( ) cos3 ( ) cos3 3 sin 3f x x x f x x x x
Example:
Example
xdx
xd 2sectan
2tansec
d u duu
dx dx
Example: )tan( 2xy
Example
xdx
xd 2sectan
2tansec
d u duu
dx dx
Example:
2 ' 2 2tan( ) 2 sec ( )y x y x x
More Trig Derivatives…
xxdx
xdcotcsc
csc
dx
duuu
dx
udcotcsc
csc
xdx
xd 2csccot
dx
duu
dx
ud 2csccot
Find the derivative.
22. 5cos 2 1y x
dy
dx
25 cos 2 1x
110 cos 2 1 sin 2 1 2x x
20cos 2 1 sin 2 1x x
10sin 4 2x
10 2cos 2 1 sin 2 1x x
10sin 2 2 1x
Find the derivative.
2 23. 2sec sin 1y x
dy
dx
2
22 sec sin 1x
2 2 2 24 sec sin 1 sec sin 1 tan sin 1 cos 1 2x x x x x
2 2 2 28 sec sin 1 tan sin 1 cos 1x x x x
Book Example
Example: Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.
5 33 4y x x pt: (2, 2)
Example: Given the function and a point:
a) Find an equation of the tangent line to the graph of f at a given point.
b) Use your calculator to graph the function and its tangent line
c) Use the derivative feature on the calculator to confirm your results.
232( ) 9f x x pt: (1, 4)