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

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**U.S.S. Alabama Mobile, Alabama**

Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

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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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**This pattern is called the**

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: Find: example:

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

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Another example: The chain rule can be used more than once. (That’s 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 derivative of x is one. derivative of outside function**

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

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**Don’t forget to use the chain rule!**

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