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3.6 The Chain Rule Photo by Vickie Kelly, 2002 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

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

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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:

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and another:

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and one more: This pattern is called the chain rule. dy/dx = 2(3x + 1) 1 3

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Chain Rule: If is the composite of and, then: 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 )

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Chain Rule: If is the composite of and, then: example: Find:

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We could also find the derivative at x = 2 this way:

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Here is a way to find the derivative by seeing layers: Differentiate the outside function, (keep the inner function unchanged...) …then multiply by the derivative of the inner function

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Another example: derivative of the outside power function derivative of the inside trig function It looks like we need to use the chain rule again!

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Another example: The chain rule can be used more than once. (Thats what makes the chain in the chain rule!)

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Each derivative formula will now include the chain rule! et cetera…

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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: derivative of outside function derivative of inside function The derivative of x is one.

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Dont forget to use the chain rule!

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