2.2 BASIC DIFFERENTIATION RULESTo Find:•Derivative using Constant Rule•Derivative using Power Rule•Derivative using Constant Multiple Rule•Derivative using Sum and Difference Rules•Derivative for Sine and Cosine

Makes sense, right?

Let’s see if we can come up with the power rule.

a. f(x) = x

b. f(x) = x2

c. f(x) = x3

Do you recognize a pattern?

2 2

0

( ) ( ) ( )'( ) lim

x

f x x f x x x xf x

x x

0

( ) ( ) ( )'( ) lim lim 1

x

f x x f x x x xf x

x x

22 2

0

2 (2 )lim lim 2

x

x x x x x x x xx

x x

2 3 23 2 3 22

0

3 3 (3 3 )lim lim 3

x

x x x x x x x x x x x xx

x x

3 3

0

( ) ( ) ( )'( ) lim lim

x

f x x f x x x xf x

x x

a. f(x) = x5

b. c.

Sometimes you need to rewrite to different form!

5( )g x x

4

1y

x

Find the following derivatives:

Ex. 3 p. 109 Find the slope of a graph.

Find the slope of f(x) = x4 when a.x = -1b.x = 0c.x = 1

3'( ) 4f x x

3'( 1) 4 1 4f

3'(0) 4 0 0f

3'(1) 4 1 4f

Ex 4 p. 109 Finding an Equation of a Tangent Line

Find the equation of the tangent line to the graph of f(x) = x3 when x = -2

To find an equation of a line, we need a point and a slope. The point we are looking at is (-2, f(-2)). In other words, find the y-value in the original function!f(-2) = (-2)3 = -8. So our point of tangency is (-2, -8)

Next we need a slope. Find the derivative & evaluate.f ‘(x) = 3x2 so find f ‘(-2) = 3٠(-2)2=12

Equation: (y – (-8)) = 12(x –(-2))So y = 12x +16 is the equation of the tangent line.

Informally, this states that constants can be factored out of the differentiation process.

2y

x 1 2

2

22 2

dy dx x

dx dx x

24( )

5

tf t

24 4 8'( ) 2

5 5 5

df t t t t

dt

Ex 5 p. 110 Using the Constant Multiple Rule

Ex 5 continued

2y x1 1

2 21 1

' 2 22

dy x x

dx x

3 2

1

2y

x

523 3

5 3 53

1 1 2 1 1'

2 2 3 33

dy x x or

dx xx

3

2

xy 3 3 3

' 1 2 2 2

dy x

dx

Ex 6 p. 111 Using Parentheses when differentiating

Original function Rewrite Differentiate Simplify

2

2

5y

x 22

5y x 32

' 25

y x 3

4'

5y

x

2

2

5y

x 22

25y x 32

' 225

y x 3

4'

25y

x

3

5

4y

x 35

4y x 25

' 34

y x215

'4

xy

3

5

4y

x 3320y x 2' 320 3y x 2' 960y x

This can be expanded to any number of functionsEx 7, p. 111 Using Sum and Difference Rules

a. 3( ) 2 5f x x x 2'( ) 3 2f x x 4

3( ) 4 32

xg x x x b.

3 2'( ) 2 12 3g x x x

Proof of derivative of sine:

0

sin( ) sin sin cos cos sin sinsin lim lim

x

d x x x x x x x xx

dx x x

cos sin sin 1 coslim

x x x x

x

0

1 cossinlim cos sinx

xxx x

x x

cos (1) sin (0) cosx x x

Last but not least, Ex 8, p112, Derivatives of sine and cosine

Function Derivative

5siny x ' 5cosy x

sin 1sin

5 5

xy x

1 cos' cos

5 5

xy x

3cosy x x ' 1 3siny x

Assign: 2.2a p. 115 #1-65 every other odd

Heads up – each of you will need to create a derivative project – something that you will use to remember all the derivative rules we learn in this chapter. This will be due Monday Oct 17. See paper for details. (online too)