3.6The Chain Rule
Jan 05, 2016
3.6 The Chain Rule
sin x2 +1⎛⎝
⎞⎠
sin 49x2 +1⎛⎝
⎞⎠
49x2 +1 7x2+7
sin x2 +1( )7x
g f x( )⎛⎝⎜
⎞⎠⎟
g h f x( )⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
h g f x( )⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
f h h x( )⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
f g h x( )⎛
⎝⎜
⎞
⎠⎟
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
( )23Find f (t) if f(t) = 3x - 7x′
( )( )( )
23
3 3
6 4 2
5 3
Find f (t) if f(t) = 3x - 7x
f (t) = 3x - 7x 3x - 7x
= 9x - 42x + 49x
f (t) = 54 x - 168 x + 98 x
′
′
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= −
( )2 3 5y x= −
If 3 5u x= −
then 2y u=
6 10y x= − 2y u= 3 5u x= −
6dy
dx= 2
dy
du= 3
du
dx=
dy dy du
dx du dx= ⋅
6 2 3= ⋅
→
and another:
5 2y u= −
where 3u t=
( )then 5 3 2y t= −
3u t=
15dy
dt= 5
dy
du= 3
du
dt=
dy dy du
dt du dt= ⋅
15 5 3= ⋅
( )5 3 2y t= −
15 2y t= −
5 2y u= −
→
and one more:29 6 1y x x= + +
( )23 1y x= +
If 3 1u x= +
3 1u x= +
18 6dy
xdx
= + 2dy
udu
= 3du
dx=
dy dy du
dx du dx= ⋅
2y u=
2then y u=
29 6 1y x x= + +
( )2 3 1dy
xdu
= +
6 2dy
xdu
= +
( )18 6 6 2 3x x+ = + ⋅This pattern is called the chain rule.
→
dy dy du
dx du dx= ⋅Chain Rule:
If is the composite of and , then:f go ( )y f u= ( )u g x=
( ) ( ) at at xu g xf g f g=′ ′ ′= ⋅o
example: ( ) sinf x x= ( ) 2 4g x x= − Find: ( ) at 2f g x′ =o
( ) cosf x x′ = ( ) 2g x x′ = ( )2 4 4 0g = − =
( ) ( )0 2f g′ ′⋅
( ) ( )cos 0 2 2⋅ ⋅
1 4⋅ 4=→
We could also do it 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 faster way to find the derivative:
( )2sin 4y x= −
( ) ( )2 2cos 4 4d
y x xdx
′ = − ⋅ −
( )2cos 4 2y x x′ = − ⋅
Differentiate the outside function...
…then the inside function
At 2, 4x y′= =
→
Another example:
( )2cos 3d
xdx
( ) 2cos 3
dx
dx⎡ ⎤⎣ ⎦
( ) ( )2 cos 3 cos 3d
x xdx⋅⎡ ⎤⎣ ⎦
derivative of theoutside function
derivative of theinside 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”!)
→
Derivative formulas 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
=
etcetera…
The formulas are written with instead of . Don’t
forget to include the term!
u′u′
du
dx
→
The most common mistake on the chapter 3 test is to forget to use the chain rule.
Every derivative problem could be thought of as a chain-rule problem:
2dx
dx2d
x xdx
= 2 1x= ⋅ 2x=
derivative of outside function
derivative of inside function
The derivative of x is one.
→
The chain rule enables us to find the slope of parametrically defined curves:
dy dy dx
dt dx dt= ⋅
dydydt
dx dxdt
=
Divide both sides bydx
dtThe slope of a parametrized curve is given by:
dydy dt
dxdxdt
=
→
These are the equations for an ellipse.
Example: 3cosx t= 2siny t=
3sindx
tdt
=− 2cosdy
tdt
=2cos
3sin
dy t
dx t=
−2
cot3
t=−
Don’t forget to use the chain rule!
( )23Find f (t) if f(t) = 3x - 7x using Chain Rule′
( )
( ) ( )
23
3 2
5 3
Find f (t) if f(t) = 3x - 7x using Chain Rule
f (t) = 2 3x - 7x 9x - 7
= 54 x - 168 x + 98 x
′
′