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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.
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Chain Rule:
If is the composite of and , then:
example: Find:
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We could also do it this way:
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Here is a faster way to find the derivative:
Differentiate the outside function...
…then the inside function
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Another example:
derivative of theoutside function
derivative of theinside 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.
(That’s what makes the “chain” in the “chain rule”!)
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Derivative formulas include the chain rule!
etcetera…
The formulas on the memorization sheet are written with
instead of . Don’t forget to include the term!
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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:
derivative of outside function
derivative of inside function
The derivative of x is one.
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The chain rule enables us to find the slope of parametrically defined curves:
Divide both sides byThe slope of a parametrized curve is given by:
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These are the equations for an ellipse.
Example:
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Don’t forget to use the chain rule!
p