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OBJECTIVE:

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

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

1.
The Power Rule

and other

Rules for Differentiation

Mr. Miehl

miehlm@tesd.net

and other

Rules for Differentiation

Mr. Miehl

miehlm@tesd.net

2.
Rules for Differentiation

Taking the derivative by using the definition is a lot of work.

Perhaps there is an easy way to find the derivative.

Taking the derivative by using the definition is a lot of work.

Perhaps there is an easy way to find the derivative.

3.
Objective

Todifferentiate functions using the

power rule, constant rule, constant

multiple rule, and sum and difference

rules.

Todifferentiate functions using the

power rule, constant rule, constant

multiple rule, and sum and difference

rules.

4.
The Derivative is …

Used to find the “slope” of a function at a point.

Used to find the “slope of the tangent line” to

the graph of a function at a point.

Used to find the “instantaneous rate of change”

of a function at a point.

Computed by finding the limit of the difference

quotient as ∆x approaches 0. (Limit Definition)

Used to find the “slope” of a function at a point.

Used to find the “slope of the tangent line” to

the graph of a function at a point.

Used to find the “instantaneous rate of change”

of a function at a point.

Computed by finding the limit of the difference

quotient as ∆x approaches 0. (Limit Definition)

5.
Notations for the

Derivative of a Function

f '( x) “f prime of x”

y' “y prime”

“the derivative of y with respect to x”

is a noun.

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

Derivative of a Function

f '( x) “f prime of x”

y' “y prime”

“the derivative of y with respect to x”

is a noun.

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

6.
Rules for Differentiation

Differentiation

is the process of

computing the derivative of a

function.

You may be asked to:

Differentiate.

Derive.

Find the derivative of…

Differentiation

is the process of

computing the derivative of a

function.

You may be asked to:

Differentiate.

Derive.

Find the derivative of…

7.
Video Clip from

The Power Rule

The Power Rule

8.
Rules for Differentiation

Working with the definition of the

derivative is important because it

helps you really understand what the

derivative means.

Working with the definition of the

derivative is important because it

helps you really understand what the

derivative means.

9.
The Power Rule

d N

[ x ] Nx N 1 , N is any real number

d

[ x] 1

dx

d N

[ x ] Nx N 1 , N is any real number

d

[ x] 1

dx

10.
The Constant Rule

d

[c] 0, c is a constant

dx

The derivative of a constant function

is zero.

d

[c] 0, c is a constant

dx

The derivative of a constant function

is zero.

11.
The Constant Multiple Rule

d

[c f ( x) ] c f '( x) , c is a constant

dx

Thederivative of a constant times a

function is equal to the constant

times the derivative of the function.

d

[c f ( x) ] c f '( x) , c is a constant

dx

Thederivative of a constant times a

function is equal to the constant

times the derivative of the function.

12.
The Sum and Difference Rules

d

[ f ( x) g ( x)] f '( x) g '( x)

dx

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

d

[ f ( x) g ( x)] f '( x) g '( x)

dx

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

d

[ f ( x) g ( x)] f '( x) g '( x)

dx

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

d

[ f ( x) g ( x)] f '( x) g '( x)

dx

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

13.
Constant Rule

Find the derivative of:

f ( x) 7

f '( x) 0

y 3

dy

0 or y ' 0

dx

Find the derivative of:

f ( x) 7

f '( x) 0

y 3

dy

0 or y ' 0

dx

14.
Power Rule

Differentiate:

f ( x) x 3 g ( x) x100

2 99

f '( x) 3 x g '( x) 100 x

y x 9

dy 8

9 x

dx

Differentiate:

f ( x) x 3 g ( x) x100

2 99

f '( x) 3 x g '( x) 100 x

y x 9

dy 8

9 x

dx

15.
Constant Multiple Rule

Find the derivative of:

1

y 2 x 3

dy 2

2 1

3 x

3

dx

dy 2

2

dx 3 x 3

Find the derivative of:

1

y 2 x 3

dy 2

2 1

3 x

3

dx

dy 2

2

dx 3 x 3

16.
Constant Multiple Rule

Find the derivative of:

4x2 4 2

f ( x) x

5 5

f '( x) 54 2x

8

f '( x) x

5

Find the derivative of:

4x2 4 2

f ( x) x

5 5

f '( x) 54 2x

8

f '( x) x

5

17.
Constant Multiple Rule

Find the derivative of:

7

g ( x) 5 x

6

g '( x) 35 x

Find the derivative of:

7

g ( x) 5 x

6

g '( x) 35 x

18.
Rewriting Before Differentiating

Function Rewrite Differentiate Simplify

5 5 5 15

f ( x) 3 f ( x) x 3 f '( x ) ( 3 x 4 ) f '( x )

2x 2 2 2 x4

Function Rewrite Differentiate Simplify

5 5 5 15

f ( x) 3 f ( x) x 3 f '( x ) ( 3 x 4 ) f '( x )

2x 2 2 2 x4

19.
Rewriting Before Differentiating

Function Rewrite Differentiate Simplify

7 7 2 7 14

g( x ) 2 g( x ) x g '( x ) (2 x ) g '( x ) x

3x 3 3 3

Function Rewrite Differentiate Simplify

7 7 2 7 14

g( x ) 2 g( x ) x g '( x ) (2 x ) g '( x ) x

3x 3 3 3

20.
Rewriting Before Differentiating

Function Rewrite Differentiate Simplify

1 1 12 1

h( x ) x h( x ) x 2 h '( x ) x h '( x ) 1

2 2x 2

Function Rewrite Differentiate Simplify

1 1 12 1

h( x ) x h( x ) x 2 h '( x ) x h '( x ) 1

2 2x 2

21.
Rewriting Before Differentiating

Function Rewrite Differentiate Simplify

1

j( x ) 2

1 2x

3

1 2 53 1

j( x ) j '( x ) x j '( x ) 5

3

2 x 2

2 3 3x 3

1 23

j( x ) x

2

Function Rewrite Differentiate Simplify

1

j( x ) 2

1 2x

3

1 2 53 1

j( x ) j '( x ) x j '( x ) 5

3

2 x 2

2 3 3x 3

1 23

j( x ) x

2

22.
Sum & Difference Rules

Differentiate:

f ( x) 5 x 2 7 x 6

f '( x) 10x 7

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

g '( x) 24x 5 15x 4 20x 5

Differentiate:

f ( x) 5 x 2 7 x 6

f '( x) 10x 7

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

g '( x) 24x 5 15x 4 20x 5

23.
Conclusion

Notations for the derivative:

f '( x) dy

y'

dx

The derivative of a constant is zero.

To find the derivative of f (x) = xN

1. Pull a copy of the exponent out in

front of the term.

2. Subtract one from the exponent.

Notations for the derivative:

f '( x) dy

y'

dx

The derivative of a constant is zero.

To find the derivative of f (x) = xN

1. Pull a copy of the exponent out in

front of the term.

2. Subtract one from the exponent.