3.6 The Chain Rule - PowerPoint Presentation

Slide1

3.6 The Chain RuleSlide2

Bell Ringer

Solve even #’sSlide3

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. Slide4

Consider a simple composite function:Slide5

and another:Slide6

and one more:

This pattern is called the

chain rule

.Slide7

Chain Rule:

If is the composite of and , then:

example:

Find:Slide8

We could also do it this way:Slide9

Here is a faster way to find the derivative:

Differentiate the outside function...

…then the inside functionSlide10

Another example:

derivative of the

outside function

derivative of the

inside function

It looks like we need to use the chain rule again!Slide11

Another example:

The chain rule can be used more than once.

(That’s what makes the “chain” in the “chain rule”!)Slide12

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!Slide13

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.Slide14

The chain rule enables us to find the slope of parametrically defined curves:

Divide both sides by

The slope of a parametrized curve is given by:Slide15

These are the equations for an ellipse.

Example:Slide16

Example:

Now we can find the slope for any value of

t

:

For example, when :Slide17

Don’t

forget to use the chain rule!pSlide18

Homework:

3.6a 3.6 p153 1,15,31,45,61

3.5 p146 21,27,33,43 2.1 p66 9,18,27,36 3.6b 3.6 p153 5,7,21,23,35,37,51,53 2.1 p66 41,44,55 3.6c 3.6 p153 9,13,27,39,43,57 2.2 p76 9,18,27,36