Post on 20-Jul-2016
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3.6 The Chain Rule
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
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
2 3 5y x
If 3 5u x
then 2y u
6 10y x 2y u 3 5u x
6dydx
2dydu
3dudx
dy dy dudx du dx
6 2 3
and another:
5 2y u
where 3u t
then 5 3 2y t
3u t
15dydt
5dydu
3dudt
dy dy dudt 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
3dudx
dy dy dudx 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 dudx du dx
Chain Rule:
If is the composite of and , then:f g y f u u g x
at at xu g xf g f g
example: sinf x x 2 4g x x Find: at 2f g x
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 dudx du dx
cos 2dy u xdx
2cos 4 2dy x xdx
2cos 2 4 2 2dydx
cos 0 4dydx
4dydx
Here is a faster way to find the derivative:
2sin 4y x
2 2cos 4 4dy 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 3d x
dx
2 cos 3 cos 3dx 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 3d x
dx
2 cos 3 cos 3dx xdx
2cos 3 sin 3 3dx 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 nudx dx
sin cosd duu udx dx
cos sind duu udx dx
2tan secd duu udx dx
etcetera…
The formulas on the memorization sheet are written with instead of . Don’t forget to include the term!
uudu
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:
2d xdx
2 dx 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 dxdt dx dt
dydydt
dx dxdt
Divide both sides bydxdtThe 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
2cos3sin
dy tdx t
2 cot3
t
Don’t forget to use the chain rule!