2.2 Basic Differentiation Rules& Rates of Change

The derivative of a constant function is 0.

That is, if is a real number, then

The Con

st

ant ul

0

R e

c

dc

dx

Quiz

Quiz2

2.2 Basic Differentiation Rules& Rates of Change

Function Derivative

7y 0dy

dx

0f x ' 0f x 3s t ' 0s t

2 , is constanty k k ' 0y

1

If is a rational number, then the function

is differentiable a

The Power

nd

.

Rule

n

n n

n

f x x

dx nx

dx

2.2 Basic Differentiation Rules& Rates of Change

Function Derivative

07y x 0dy

dx

y x ' 1y 3s t t 2' 3s t t

3g x x 2 / 32 / 3

1 1'

3 3g x

x

2

1y

x 2 3

3

22

dy dx x

dx dx x

2.2 Basic Differentiation Rules& Rates of Change

4

3

Find the slope of when

a 1 b 0 c 1

' 4

' 1

' 0

' 1

4

0

4

f x x

x x x

f x x

f

f

f

p. 104

2

tangent

1 1

Find an equation of the tangent line to the graph

of when 2.

' 2 ' 2 4 4

2 4 the tangent goes through 2,4 .

4 4 2

f x x x

f x x f

y

y y m

m

x

f

x x

p. 104

AP ExamMultiple Choice

If is a differentiable function and

is a real number, then is also

differentiable and

The Constant Multip

'

le Rule

.

f

c cf

dcf x cf x

dx

2.2 Basic Differentiation Rules& Rates of Change

Function Derivative

2 /y x 22 /dy

xdx

24

5

tf t 8

'5

f t t

2y x1dy

dx x

3 2

1

2y

x

5 / 3

1

3

dy

dx x

3

2

xy

3'

2y

Function Rewrite Differentiate Simplify

3

5

2y

x 35

2y x

415'

2y x

4

15'

2y

x

3

5

2y

x 35

8y x

415'

8y x

4

15'

8y

x

2

7

3y

x 27

3y x

14'

3y x

2

7

3y

x 263y x ' 126y x

2.2 Basic Differentiation Rules& Rates of Change

The derivative of the sum or difference of two

differentiable functions is differentiable and is the

sum or difference of their derivative

The Sum and Differe

Sum Rule

nce Rules

s.

' '

df x g x f x g x

dxd

' Difference Rule' f x g x f x g xdx

2.2 Basic Differentiation Rules& Rates of Change

Function Derivative

3 4 5f x x x 2' 3 4f x x

4

33 22

xg x x x 3 2' 2 9 2g x x x

2.2 Basic Differentiation Rules& Rates of Change

Derivatives of Sine and Cosine Functio

sin cos

ns

cos sind d

x x x xdx dx

2.2 Basic Differentiation Rules& Rates of Change

Function Derivative

2sinf x x ' 2cosf x xsin

2

xy

1' cos

2y x

2.2 Basic Differentiation Rules& Rates of Change

cosy x x ' 1 siny x

Change in Distance

Change in TimeAverage Velocity

s

t

2

If a billiard ball is dropped from a height of 100 feet, its height

at time is given by

16 100

where is measured in feet and is measured in seconds. Fi

Position Function

s t

s t

s t

nd

the average velocity over each of the following time intervals.

(a) [1,2] (b) [1,1.5] (c) [1,1.1] 2 1 36 84

48 feet per seco1

n1

d2

s ss

t

1.5 1 20

1.5 1 0.40 feet per secon

5d

s ss

t

1.1 1 80.64 84

33.6 feet pe1

r .1 1 0

secon1

d.

s ss

t

Instantaneous Velocity

p. 109

1 2

1

The average velocity between and is the slope

of the secant line, and the instantaneous velocity at

is the slope of the tangent line.

t t

t Avg. Velocity

p. 109

2

At time 0, a diver jumps from a diving board that is 32 feet

above the water. The position of the diver is given by

where is measured in feet and is

16 16

measured in s

32

econ

t

s

s t

t

t t

ds.

(a) When does the diver hit the water?

(b) What is the diver's velocity at impact?

2

2

16 16 32 0

16 2 0

16 2 1 0

2

48 feet per second

' 32 16

' 2

s t t t

t t

t t

s t

t

t

s

HW 2.2/1,3,11,15,17,19,21,24,26,28,29,33,35,37,41,45,46,49

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Mr. Gibbons: as himself

Special thanks to:Bob’s fish parlor

The Communist partyThe Clinton administration

Nintendo of America

No thanks to:Fred’s fish parlorThe Nazi party

The Bush administrationNintendo of Japan

And a very special thanks to Mr. Gibbons for teaching us these crazy derivatives

Sketch the graph of a function whose

derivative is always negative.

Lesson

3Find the derivative of 2 4.s t t t

2' 3 2s t t

Lesson