3.3

Rules for Differentiation

What you’ll learn about Positive Integer Powers, Multiples,

Sums and Differences Products and Quotients Negative Integer Powers of x Second and Higher Order Derivatives

… and whyThese rules help us find derivatives of functions analytically

in a more efficient way.

Rule 1 Derivative of a Constant Function

If is the function with the constant value , then

0

This means that the derivative of every constant function

is the zero function.

f c

df dc

dx dx

Rule 2 Power Rule for Positive Integer Powers of x.

1

If is a positive integer, then

The Power Rule says:

To differentiate , multiply by and subtract 1 from the exponent.

n n

n

n

dx nx

dx

x n

Rule 3 The Constant Multiple Rule

If is a differentiable function of and is a constant, then

This says that if a differentiable function is multiplied by a constant,

then its derivative is multiplied by the same cons

u x c

d ducu c

dx dx

tant.

Rule 4 The Sum and Difference Rule

If and are differentiable functions of , then their sum and differences

are differentiable at every point where and are differentiable. At such points,

.

u v x

u v

d du dvu v

dx dx dx

Example Positive Integer Powers, Multiples, Sums, and Differences 4 2 3

Differentiate the polynomial 2 194

That is, find .

y x x x

dy

dx

4 2Sum and Difference Rule

3Constant and Power Rules

3

By Rule 4 we can differentiate the polynomial term-by-term,

applying Rules 1 through 3.

32 19

4

34 2 2 0

43

= 4 44

dy d d d dx x x

dx dx dx dx dx

x x

x x

Example Positive Integer Powers, Multiples, Sums, and Differences

4 2Does the curve 8 2 have any horizontal tangents?

If so, where do they occur?

Verify you result by graphing the function.

y x x

4 2 3

If any horizontal tangents exist, they will occur where the slope

is equal to zero. To find these points we will set 0 and solve for .

Calculate 8 2 4 16

Set 0 and solve

dy

dxdy

xdx

dy dx x x x

dx dxdy

dx

3

2 2

for

4 16 0

4 4 0; 4 0 4 0

This gives horizontal tangents at 0, 2, 2.

x

x x

x x x x

x

Rule 5 The Product Rule

The product of two differentiable functions and is differentiable, and

The derivative of a product is actually the sum of two products.

u v

d dv duuv u v

dx dx dx

Example Using the Product Rule 3 2Find if 4 3f x f x x x

3 2

3 2 3 2 2

4 4 2

4 2

Using the Product Rule with 4 and 3,gives

4 3 4 2 3 3

2 8 3 9

5 9 8

u x v x

df x x x x x x x

dx

x x x x

x x x

Rule 6 The Quotient Rule

2

At a point where 0, the quotient of two differentiable

functions is differentiable, and

Since order is important in subtraction, be sure to set up the

numerator of the

uv y

v

du dvv ud u dx dx

dx v v

Quotient rule correctly.

Example Using the Quotient Rule

3

2

4Find if

3

xf x f x

x

3 2

3 2 2 3

22 2

4 2 4

22

4 2

22

Using the Quotient Rule with 4 and 3,gives

4 3 3 4 2

3 3

3 9 2 8

3

9 8

3

u x v x

x x x x xdf x

dx x x

x x x x

x

x x x

x

Rule 7 Power Rule for Negative Integer Powers of x

1

If is a negative integer and 0, then

.

This is basically the same as Rule 2 except now is negative.

n n

n x

dx nx

dxn

Example Negative Integer Powers of x

1Find an equation for the line tangent to the curve at the point 1,1 .y

x

1

22

Rewrite the function as and use the Power Rule to

find the derivative.

11

1Evaluate 1 = 1

1The line through 1,1 with slope 1 is

1 1 1

2

This shows the graph of the funct

y x

y xx

y

m

y x

y x

ion and its tangent line at (1, 1).

1yx

2y x

Second and Higher Order Derivatives

2

2

The derivative is called the of with respect to .

The first derivative may itself be a differentiable function of . If so,

its derivative, ,

dyy first derivative y x

dxx

dy d dy d yy

dx dx dx dx

3

3

is called the of with respect to . If

double prime is differentiable, its derivative,

,

is called the of with respect to .

second derivative y x y

y

dy d yy

dx dxthird derivative y x

Second and Higher Order Derivatives

1

The multiple-prime notation begins to lose its usefulness after three primes.

So we use " super "

to denote the th derivative of with respect to .

Do not confuse the notation with th

n n

n

dy y y n

dxn y x

y

e th power of , which is . nn y y

Quick Quiz Sections 3.1 – 3.3

You may use a graphing calculator to solve the following problems.

1. Let 1 . Which of the following statements about are true?

I. is continuous at 1.

II. is differentiable at 1.

III. has

f x x f

f x

f x

f

a corner at 1.

A I only

B II only

C III only

D I and III only

x

Quick Quiz Sections 3.1 – 3.3

You may use a graphing calculator to solve the following problems.

1. Let 1 . Which of the following statements about are true?

I. is continuous at 1.

II. is differentiable at 1.

III. has

f x x f

f x

f x

f

a corner at 1.

A I only

B II only

D

C III only

I and III only

x

Quick Quiz Sections 3.1 – 3.3

2. If the line normal to the graph of at the point 1,2 passes through

the point 1,1 , then which of the following gives the value of 1 ?

A 2

B 2

1C

21

D 2

E 3

f

f

Quick Quiz Sections 3.1 – 3.3

2. If the line normal to the graph of at the point 1,2 passes through

the point 1,1 , then which of the following gives the value of 1 ?

A 2

B 2

1C

12

E

D

32

f

f

Quick Quiz Sections 3.1 – 3.3

2

2

2

2

4 33. Find if .

2 110

A4 3

10B

4 3

10C

2 1

10D

2 1

E 2

dy xy

dx x

x

x

x

x

Quick Quiz Sections 3.1 – 3.3

2

2

2

2

1

4 33. Find if .

2 1

10B

4 3

10C

2 1

10D

2 1

E

0A

4

2

3x

dy xy

dx x

x

x

x