AP CalculusChapter 2, Section 2

Basic Differentiation Rules and Rates of Change

2013 - 2014

The Constant Rule

• The derivative of a constant function is 0. That is, if c is a real number, then

Find the derivative of each functionFunction Derivative

Exploration: The Power Rule

• Use the definition of a derivative to find the derivative of each function. Do you see the pattern??

Function Derivative

The Power Rule

• If n is a rational number, then the function is differentiable and

Using the Power RuleFunction Derivative

Find the slope of a graph

• Find the slope of the graph of when

Find an equation of the tangent line to the graph of when

The Constant Multiple Rule

• If f is a differentiable function and c is a real number, then cf is also differentiable and .

Using the Constant Multiple RuleFunction Derivative

Using Parentheses When Differentiating

Original Function

Rewrite Differentiate Simplify

The Sum and Difference Rules

• The sum (or difference) of two differentiable functions f and g is itself differentiable.

• The derivative of f + g or (f - g) is the sum (or difference) of the derivatives of f and g.

Using the Sum and Difference Rules

Function Derivative

Derivatives of Sine and Cosine Functions

• Using the limits you memorized from Chapter 1, you can prove the following derivatives.

Derivatives Involving Sines and Cosines

Function Derivative

Rates of Change• Derivates are also used to determine rates of change from one

variable with respect to another.

• Common use for rates of change is when an object is moving in a straight line.

• The function s that gives the position (relative to the origin) of an object as a function of time t is called a position function.

• Over a period of time ∆t, the object changes its position by the amount .

• Then, using the formula the average velocity is

Finding Average Velocity of a Falling Object

• If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function

Where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals.a) [1, 2]b) [1, 1.5]c) [1, 1.1]

Suppose in the billiard ball problem that you wanted to find the instantaneous velocity of the

object when . This would involve the derivative of the function. The derivative finds the instantaneous velocity at a given point since would be approaching 0. The speed of an object would be the absolute

value of the velocity since speed cannot be negative, but velocity can.

Using the Derivative to Find Velocity

• At time , a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by where s is measured in feet and t is measured in seconds.

• a) When does the diver hit the water?• B) What is the diver’s velocity at impact?

Ch. 2.2 Homework

• Pg. 115 – 117: 3 – 23 odd, 27, 31, 33, 39, 43, 45, 51, 53, 57, 83, 85, 93, 95, 103

• Total problems: 25