Basic Differentiation Rules and Rates of Change

Section 2.2

The Constant Rule

][cdx

d

Examples:

1. y = 5, find (dy/dx)

2. f(x) = 13, find f’(x)

3. y = (kπ)/2 where k is an integer, find y’

The Power Rule

][,)( nn xdx

dxxf

Examples:

1. f(x) = x4, find f’(x)

2. g(x) = , find g’(x)

3. , find

4 x

3

1

xy

dx

dy

The Constant Multiple Rule

)]([ xcfdx

d

Examples:

Function Derivative y = 3/x f(t) = (3t2)/7 y = 5 y =

y = (-5x)/3

x

4 33

1

x

Example:

Find the slope of the graph of f(x) = x7 when

a. x = -2 b. x = 5

Example:

Find the equation of the tangent line to the graph of f(x) = x3, when x = -3

The Sum and Difference Rules Sum Rule

Difference Rule

Example: f(x) =

Find f’(x)

1426

23

xxx

Derivatives of Sine and Cosine

Examples:

1. y = 3 cos x

2. y =

3. y = 2x2 + cos x

][sin xdx

d ][cos xdx

d

5

sin x

Extra Examples:

Find each derivative:

x

xx 132 2

3

)1( 2 xx

Rates of Change: Average Velocity Velocity (Rate) =

If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function s = -16t2 + 100.

Find the average velocity over each time interval: a. [1, 2] b. [1, 1.5] c. [1,1.1]

Instantaneous Velocity:If s(t) is the position function, the velocity of an

object is given by .Velocity can be .Speed is the of the velocity.Example: At time t = 0, a diver jumps from a

board that is 32 feet above the water. The position of the diver is given by

s(t) = -16t2+16t + 32a. When does the diver hit the water?b. What is the diver’s velocity at impact?