Rules for DifferentiationRules for Differentiation

Rules for DifferentiationRules for DifferentiationTaking the derivative by using the definition is a lot of work.

Perhaps there is an easy way to find the derivative.

ObjectiveObjective

To differentiate functions using the To differentiate functions using the power rule, constant rule, constant power rule, constant rule, constant multiple rule, and sum and difference multiple rule, and sum and difference rules.rules.

TS: Making decisions after reflection TS: Making decisions after reflection and review.and review.

The Derivative is …The Derivative is … Used to find the slope of a function at a point.Used to find the slope of a function at a point.

Used to find the slope of the tangent line to the Used to find the slope of the tangent line to the graph of a function at a point. graph of a function at a point.

Used to find the instantaneous rate of change of Used to find the instantaneous rate of change of a function at a point. a function at a point.

Computed by finding the limit of the difference Computed by finding the limit of the difference quotient as ∆x approaches 0. (Limit Definition) quotient as ∆x approaches 0. (Limit Definition)

Notations for the Notations for the Derivative of a FunctionDerivative of a Function

d

dx

'( )f x

'y

dy

dx

dy

dx

““ff prime of prime of xx””

““yy prime” prime”

““the derivative of the derivative of yy with respect to with respect to xx””

is a verb. “Take a derivative the respect to is a verb. “Take a derivative the respect to xx…” …”

is a noun.is a noun.

Rules for DifferentiationRules for Differentiation

DifferentiationDifferentiation: the process of : the process of computing the derivative of a computing the derivative of a function.function.

You may be asked to:You may be asked to: Differentiate.Differentiate. Derive.Derive. Find the derivative of…Find the derivative of…

Video Clip fromVideo Clip fromCalculus-Help.comCalculus-Help.com

The Power RuleThe Power Rule

Rules for DifferentiationRules for Differentiation

Working with the definition of the Working with the definition of the derivative is important because it derivative is important because it helps you really understand what the helps you really understand what the derivative means, and it can be used derivative means, and it can be used with all kinds of functions.with all kinds of functions.

The Power Rule The Power Rule

1[ ] , is any real numberN Ndx Nx N

dx

1 0[ ] 1 1d d

x x xdx dx

Be careful if you are taking the power of x, or 2x, or 3x, etc.

The Constant Rule The Constant Rule

The derivative of a constant function is The derivative of a constant function is zero.zero.

[ ] 0, is a constantd

c cdx

0 1[ ] 0 0, is a constantd d

c x c x c cdx dx

The Constant Multiple Rule The Constant Multiple Rule

[ ( ) ] '( ) , is a constantd

c f x c f x cdx

The derivative of a constant times a The derivative of a constant times a function is equal to the constant function is equal to the constant times the derivative of the function.times the derivative of the function.

The Sum and Difference Rules The Sum and Difference Rules

[ ( ) ( )] '( ) '( )d

f x g x f x g xdx

[ ( ) ( )] '( ) '( )d

f x g x f x g xdx

The derivative of a sum is the sum of the derivatives.

The derivative of a difference is the difference of the derivatives.

Constant RuleConstant Rule

Find the derivative of:Find the derivative of:( ) 7f x '( ) 0f x

3y

0dy

dx or ' 0y

Power RulePower Rule

Differentiate:Differentiate:3( )f x x

2'( ) 3f x x

9y x

89dy

xdx

100( )g x x99'( ) 100g x x

Constant Multiple RuleConstant Multiple Rule

Find the derivative of:Find the derivative of:132y x

2 dy

dx

231

3 x

23

2

3

dy

dx x

Constant Multiple RuleConstant Multiple Rule

Find the derivative of:Find the derivative of:24

( )5

xf x

45'( ) f x 2x

8'( )

5f x x

24

5x

Constant Multiple RuleConstant Multiple Rule

Find the derivative of:Find the derivative of:7( ) 5g x x

6'( ) 35g x x

Rewriting Before DifferentiatingRewriting Before Differentiating

FunctionFunction RewriteRewrite DifferentiateDifferentiate SimplifySimplify

3

5( )

2f x

x 35

( )2

f x x 45'( ) ( 3 )

2f x x 4

15'( )

2f x

x

Rewriting Before DifferentiatingRewriting Before Differentiating

FunctionFunction RewriteRewrite DifferentiateDifferentiate SimplifySimplify

2

7( )

3g x

x 27( )

3g x x

7'( ) (2 )

3g x x

14'( )

3g x x

Rewriting Before DifferentiatingRewriting Before Differentiating

FunctionFunction RewriteRewrite DifferentiateDifferentiate SimplifySimplify

( )h x x12( )h x x

12

1'( )

2h x x

12

1'( )

2h x

x

Rewriting Before DifferentiatingRewriting Before Differentiating

FunctionFunction RewriteRewrite DifferentiateDifferentiate SimplifySimplify

23

1( )

2j x

x

23

1( )

2j x

x

53

1 2'( )

2 3j x x

53

1'( )

3j x

x

23

1( )

2j x x

Sum & Difference RulesSum & Difference Rules

Differentiate:Differentiate:2( ) 5 7 6f x x x

'( )f x

6 5 2( ) 4 3 10 5 16g x x x x x

'( )g x

10x 7

524x 415x 20x 5

ConclusionConclusion

Notations for the derivative:Notations for the derivative:

The derivative of a constant is zero.The derivative of a constant is zero. To find the derivative of To find the derivative of f f ((xx) = ) = xxNN

1.1. Pull a copy of the exponent out in Pull a copy of the exponent out in front of the term.front of the term.

2.2. Subtract one from the exponent.Subtract one from the exponent.

'( )f x 'y dy

dx