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Basic Differentiation Rules and Rates of Change

- Section 2.2

After this lesson, you should be able to

- Find the derivative using the Constant Rule.
- Find the derivative using the Power Rule.
- Find the derivative using the Constant Multiple

Rule and the Sum and Difference Rules. - Find the derivative of sine and cosine.
- Use derivatives to find rates of change.

Slope of Tangent Line at (x, f(x))

Q

P

Let

Rules For Computing Derivatives

Constant Rule

Power Rule

Ex

Ex

Rules For Computing Derivatives

Ex

Ex

Rules For Computing Derivatives

Examples

Example

Examples

Example Find the derivative of

Examples

Example Find the derivative of

Finding the Equation of a Tangent Line

Example Find the equation of the tangent line

at x1 and take a peek at the graph.

Pt f(1)

(1,2)

peek at the graph

slope f(x)

5x4 6x

f(1)

-1

equation

y 2 (-1)(x 1)

y -x 3

Finding an Equation of a Horizontal Tangent Line

Example Find an equation for the horizontal

tangent line to

peek at the graph

Rates of Change

Some Examples population growth rates,

production rates, velocity and acceleration

Common example is the motion of an object along a

straight line.

Motion Along a Straight Line

s position

S(t) position function

S (ft)

S(tt)

Ave. velocity during the time interval from t to

(tt)

S(t)

t (sec)

t

(tt)

Instantaneous velocity at time t

Motion Along a Straight Line

v(t) gt 0,

v(t) lt 0,

v(t) 0,

stopped instantaneously

rate of change of position

s(t) position (ft)

rate of change of velocity

v(t) s(t) velocity (ft/sec)

a(t) v(t) s(t) accel (ft/s2)

speed v(t)

Free Falling Object

gravity

initial velocity

initial position

Example of Free Falling Object

Example A ball is thrown straight down from the

top of a 220-foot building with an initial

velocity of 22 feet/second. What is its

velocity after 3 seconds What is its velocity

after falling 108 feet

220

Describes the motion of the object

0

Example of Free Falling Object

Example A ball is thrown straight down from the

top of a 220-foot building with an initial

velocity of 22 feet/second. What is its

velocity after 3 seconds What is its velocity

after falling 108 feet

220

Describes the motion of the object

0

Homework

Section 2.2 page 113 1-51 odd, 53ab, 55ab, 57,

61, 81-86 all

Rate of change problems 92 and 93

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