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
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)