2.4 The Chain RuleObjective: Find the derivative of a composite function

using the Chain Rule

Ms. BattagliaAB/BC Calculus

The Chain RuleThe chain rule is used to compute the derivative of the composition of two or more functions.

Theorem 2.10 The Chain RuleIf y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x then

y=f(g(x)) u=g(x) y=f(u)

u=x+1

y=sin2x

Decomposition of a Composite Function

Using the Chain Rule

Differentiate

Differentiating Functions Involving RadicalsFind all points on the graph of for which f’(x)=0 and those for which f’(x) does not exist.

Differentiating Quotients with Constant Numerators

Simplify by Factoring Out the Least Powers

Simplifying the Derivative of a Quotient

Simplifying the Derivative of a Power

Trig Functions and the Chain Rule

a. y=sin2x b. y=cos(x-1) c. y=tan3x

Applying the Chain Rule to Trig Functions

Parenthesis & Trig Functions

A. y=cos3x B. y=(cos3x)2 C. y=cos(3x)2

Parenthesis & Trig Functions

Repeated Application of the Chain Rule

Find the equation of the tangent line to the graph of

at the point (π,1). Then determine all values of x in the interval (0,2π) at which the graph of f has a horizontal tangent.

Tangent Line of a Trig Function

Change in distanceChange in time

Ex: A billiard ball is dropped from a height of 100 ft, its height s at time t is given by the position function s = -16t2 + 100 where s is measured in feet and t is measured in seconds. Find the average veolcity over the time interval [1,2]. Find the instantaneous velocity at t=1.

Average velocity

Read 2.4, Page 137 #7-20, 45-50, 55-65 odd, 109-112

Classwork/Homework