2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

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

HWQ Let f(x) and g(x) be 2 differentiable functions such that: xF(x)G(x)F’(x)G’(x) Find the derivative of f(x)g(x) at x =

Warm-Up Evaluate the following limit:

Calculus Warm-Up

We will come back to this problem later.

The Chain Rule Copyright © Cengage Learning. All rights reserved. 2.4

Find the derivative of a composite function using the Chain Rule. Objective:

Example 1: The length, L, in cm, of a steel bar depends on the air temperature, H °C, and the temperature H depends on time, t, measured in hours. If the length increases by 2 cm for every degree increase in temperature, and the temperature is increasing at 3 °C per hour, how fast is the length of the bar increasing? What are the units for your answer?

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. We do this with the chain rule.

Consider a simple composite function:

and another:

This pattern is called the chain rule.

Chain Rule: Example: Find: or:

Chain Rule:

Differentiate the outside function... …then the inside function Chain Rule:

Use the chain rule to differentiate:

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

con’t: 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… If formulas on a memorization sheet are written with instead of. Don’t forget to include the term!

The most common mistake on derivative tests 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 by The slope of a parametrized curve is given by:

These are the equations for an ellipse. Example:

Practice: Differentiate:

Practice: Differentiate:

BC Homework 2.4 Day 1 p. 137: 7-31 odd, odd, odd, 81, Day 2: MMM pgs Day 3: MMM pg. 50

AB Homework 2.4 Day 1 p. 137: 1-31 odd, odd 2.4 Day 2: p. 137: odd, odd 2.4 Day 3: MMM pgs. 44 & Day 4: MMM pgs. 46 & 50

2.4 The Chain Rule – Day 2 Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002

HWQ Differentiate:

2.4 Warm-up

Common Denominator

HWQ (no calculator) Determine the point(s) at which the graph of has a horizontal tangent.

AB Homework 2.4 Day 1 p. 137: 1-31 odd, odd 2.4 Day 2: p. 137: odd, odd 2.4 Day 3: MMM pgs Day 4: Chain Rule W/S