15 b the chain rule. we now have a small list of “shortcuts” to find derivatives of simple...
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15BThe Chain Rule
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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.
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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
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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.
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2( ) 3f x x
23u x Inside Function
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2( ) 3f x x
23u x Inside Function
y u Outside Function
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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).
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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
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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
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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
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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
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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
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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
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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
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Next, simplify
22 36' xxxf
xxxf 233'22
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Find the derivative2 71) ( ) (3 5 )f x x x
2 232) ( ) ( 1)f x x
2
73) ( )
(2 3)f t
t
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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
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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
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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
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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:
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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
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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
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One Last Thought
It takes a big man to cry, but it takes a
bigger man to laugh at that man.
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Homework
Page 364 (#1 – 2)
Page 366 (#1 – 6)