# Calc03_6 - 3.6 The Chain Rule Greg Kelly Hanford High...

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3.6 The Chain Rule Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

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Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002 U.S.S. Alabama Mobile, 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.

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Consider a simple composite function: 6 10 y x = - ( 29 2 3 5 y x = - If 3 5 u x = - then 2 y u = 6 10 y x = - 2 y u = 3 5 u x = - 6 dy dx = 2 dy du = 3 du dx = dy dy du dx du dx = 6 2 3 =
and another: 5 2 y u = - where 3 u t = ( 29 then 5 3 2 y t = - 3 u t = 15 dy dt = 5 dy du = 3 du dt = dy dy du dt du dt = 15 5 3 = ( 29 5 3 2 y t = - 15 2 y t = - 5 2 y u = -

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and one more: 2 9 6 1 y x x = + + ( 29 2 3 1 y x = + If 3 1 u x = + 3 1 u x = + 18 6 dy x dx = + 2 dy u du = 3 du dx = dy dy du dx du dx = 2 y u = 2 then y u = 2 9 6 1 y x x = + + ( 29 2 3 1 dy x du = + 6 2 dy x du = + ( 29 18 6 6 2 3 x x + = + This pattern is called the chain rule .
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