2-2: differentiation rules objectives: learn basic differentiation rules explore relationship...

2-2: Differentiation RulesObjectives:•Learn basic differentiation rules•Explore relationship between derivatives and rates of change

© 2002 Roy L. Gover (www.mrgover.com)

Basic Differentiation Rules

1.Constant Rule

The derivative of a constant function is 0. There is no rate of change.

[ ]d

c odx

Let’s prove it.

Duh…

Find the following derivatives using the definition

𝑓 (𝑥 )=𝑥𝑓 (𝑥 )=𝑥2

𝑓 (𝑥 )=𝑥3Make a guess about

Find the following derivatives using the definitionMake a guess about

Basic Differentiation Rules

1[ ]n ndax nax

dx

2. Power Rule

Where a & n are real numbers and n is rational

Warm-UpFind the derivative using the definition (sometimes called the limit of the difference quotient) and the power rule. Confirm that you get the same answer.

2( ) 2f x x

Basic Differentiation Rules

[ ( )] ( )d d

cf x c f xdx dx

3. Constant Multiple Rule

Where c is a constant

Basic Differentiation Rules

[ ( )] '( )d

cf x cf xdx

3. Constant Multiple Rule

Note: alternate notation

( )d

f xdx

'( )f xis the same as

Basic Differentiation Rules

3. Sum or Difference Rule

( ) ( ) '( ) '( )d

f x g x f x g xdx

The derivative of the sum or difference is the sum or difference of the derivatives

Basic Differentiation Rules

4. Derivatives of Sine & Cosine

sin cosd

x xdx

cos sind

x xdx

Prove it! Now!

4. Derivatives of Sine & Cosine

sin cosd

x xdx

cos sind

x xdx

ExampleFind the derivative, if it exists:

1. 3

2. ( ) 2

3. ( ) 2

y

f x

h t

We used what rule(s)?

ExampleFind the derivative, if it exists:

2

5

1.

2. ( )

3. ( ) 3

y x

f x x

h t t

We used what rule(s)?

ExampleFind the derivative, if it exists:

3 2

11.

2. ( )

yx

f x x

We used what rule(s)?

ExampleFind the derivative, if it exists:

3

31.

25

2. ( )2

xy

f xx

We used what rule(s)?

Try ThisFind the derivative:

2

1 y

x

33

22

dyx

dx x

First write as then use the power rule:

2x

Try ThisFind the derivative:

3

2

dy x

dx

3 ( )f x xFirst write as_____, then…

Try ThisFind the derivative:

27(9 ) 126dy

x xdx

2

7 ( )

(3 )f x

x First write as_____, then…

Example Find the slope of

3( )f x xat x=

x=2

2-1

x=-1

Review

1 1( )y y m x x

Point-Slope Form:

1 at dy

x xdx

1( )f x

Review

y mx b

Slope-y intercept Form:

Example

x=2

Find the equation of line tangent to at x=2

2( )f x x

Try This Find the equation of line tangent to at x=1

3( )f x x

x=13 2y x

ExampleFind the derivative, if it exists:

3( ) 4 5f x x x

We used what rule(s)?

Try ThisFind the derivative, if it exists:

43( ) 3 2 2

2

xh x x x

3 2'( ) 2 9 2h x x x

ExampleFind the derivative, if it exists:

sin1.

2

2. ( ) 2cos

xy

f

Important Idea

The slope of the sin function at a point is the value of the cos function at the point

3

ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t

1. Find the average velocity between 1 and 3 seconds.2. Find the instantaneous velocity at 3 seconds.

ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t

1. Find the average velocity between 1 and 3 seconds.

ExampleIf an object is dropped, its height above the ground is given by 2( ) 16 200s t t 2. Find the instantaneous velocity at 3 seconds.

Lesson Close

This lesson demonstrated the use of several differentiation rules. There will be others in future lessons. You must memorize these rules.

Assignment

123/1-47 odd & 71