3.6 the chain rule photo by vickie kelly, 2002 created by greg kelly, hanford high school, richland,...
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3.6 The Chain Rule
Photo by Vickie Kelly, 2002Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
U.S.S. AlabamaMobile, Alabama
We now have a pretty good 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.
Consider a simple composite function:
6 10y x
3 52y x
nd ,5a 3u x 2( )y u
6 10y x 2( )y u 3 5u x
6dy
dx 2
dy
du 3
du
dx
dy
d
u
u
dy d
dx dx
6 32
( '( ) ( ) *')d
f g x g x xfx
gd
and another:
5 2y u
if 3u t
5 3 2( )y t
3u t
15dy
dt 5
dy
du 3
du
dt
15
5 3 2y t
15 2y t
5 2y u
dy
d
u
u
dy d
dx dx
( '( ) ( ) *')d
f g x g x xfx
gd
5 3
and one more:29 6 1y x x
23 1y x
3 1u x
3 1u x
18 6dy
xdx
2dy
udu
3du
dx
dy dy du
dx du dx
2y u
2when y u
29 6 1y x x
2 3 1dy
xdu
This pattern is called the chain rule.
dy/dx = 2(3x + 1)1 • 3
or ( ( )) ' ( ) * 'd
f g x f g x g xdx
dy dy du
dx du dx Chain Rule:
If is the composite of and , then:
f g y f u u g x
at at g x xf g f g
If f(g(x)) is the composite of y = f(u) and u = g(x), then:
d/dx(f(g(x)) = d/dx f (at g(x)) • d/dx g(at x)
dy dy du
dx du dx Chain Rule:
If is the composite of and , then:
f g y f u u g x
at at ( ) ( )( )xg x g xf g f g f g x
example:
sinf x x
Find:
cosf x x
2g x x 2 22g 0 2f g f g
1 4 4f g
' cos 10 (0)f
22 2 4 0g 2 4g x x
( )
2
g xf g f g
a x
x
t
( 22)gf g f g
We could also find the derivative at x = 2 this way:
2sin 4f g x x
2sin 4y x
siny u 2 4u x
cosdy
udu
2du
xdx
dy dy du
dx du dx
cos 2dy
u xdx
2cos 4 2dy
x xdx
2cos 2 4 2 2dy
dx
cos 0 4dy
dx
4dy
dx
Here is a way to find the derivative by seeing “layers:”
2sin 4y x
2 2' cos 4 4d
y x xdx
2' cos 4 2y x x
Differentiate the outside function,(keep the inner function unchanged...)
…then multiply by the derivative of the inner function
Evaluate this general derivative at 2, to find ' 4x y
Another example:
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
derivative of theoutside power function
derivative of theinside trig function
It looks like we need to use the chain rule again!
Another example:
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
2cos 3 sin 3 3d
x x xdx
2cos 3 sin 3 3x x
6cos 3 sin 3x x
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)
Each derivative formula will now include the chain rule!
1n nd duu nu
dx dx sin cos
d duu u
dx dx
cos sind du
u udx dx
2tan secd du
u udx dx
et cetera…
The most common mistake in differentiating is to forget to use the chain rule.
Every derivative problem could be thought of as a chain-rule situation:
2dx
dx2
dx x
dx 2 1x 2x
derivative of outside function
derivative of inside function
The derivative of x is one.
Don’t forget to use the chain rule!
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