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:

and another:

and one more:

This pattern is called the chain rule.

Chain Rule:

If is the composite of and , then:

example: Find:

We could also do it this way:

Here is a faster way to find the derivative:

Differentiate the outside function...

…then the inside function

Another example:

derivative of theoutside function

derivative of theinside function

It looks like we need to use the chain rule again!

Another example:

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!

etcetera…

The formulas on the memorization sheet are written with

instead of . Don’t forget to include the term!

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.

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:

These are the equations for an ellipse.

Example:

Don’t forget to use the chain rule!

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