Section 2.3 – The Product and Quotient Rules and Higher-Order

Derivatives

The Product RuleThe derivative of a product of functions is NOT the product

of the derivatives.

If f and g are both differentiable, then

In other words, the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

ddx f x g x f x g' x g x f ' x

Another way to write the Rule: ' 'd

u v u v v udx

Example 1Differentiate the function:

f ' x ddx 3x 2 1 7 2x 3

3x 2 1 ddx 7 d

dx 2x 3 7 2x 3 ddx 3x 2 d

dx 1

3x 2 1 0 6x 2 7 2x 3 6x 0

18x 4 6x 2 42x 12x 4

Product Rule

Sum/Difference Rule

Power Rule

Simplify

f x 3x 2 1 7 2x 3

3x 2 1 ddx 7 2x 3 7 2x 3 d

dx 3x 2 1

u v

30x 4 6x 2 42x

u v' v u'

Example 2If h(x) = xg(x) and it is known that g(3) = 5 and g'(3) =2, find h'(3).

h' x ddx xg x

xg' x g x 1

xg' x g x

h' 3 3g' 3 g 3

Product Rule

Find the derivative:

Evaluate the derivative:

x ddx g x g x d

dx x

u v

32 5

u v' u v'

11

The Quotient RuleThe derivative of a quotient of functions is NOT the quotient

of the derivatives.

If f and g are both differentiable, then

In other words, the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

ddx

f x g x g x f ' x f x g' x

g x 2

Another way to write the Rule:2

' 'd u vu uv

dx v v

“Lo-d-Hi minus Hi-d-Lo”

Example 1Differentiate the function:

24 73

' d xdx x

f x

2

22

3 4 4 7 2

3

x x x

x

2 2

22

12 4 8 14

3

x x x

x

2

22

4 14 12

3

x x

x

Quotient Rule

f x 4 x 73 x 2

2 2

22

3 4 7 4 7 3

3

d ddx dxx x x x

x

uv

u v'v u'

v2

Example 2Find an equation of the tangent line to the curve

at the point (1,½).

y x

1x 2

y'1x 2 d

dx x1 2 x ddx 1x 2

1x 2 2

1x 2 1

2 x 1 2 x 2x

1x 2 2

1x 2 1

2 x 2x x

1x 2 2

2 x

2 x

1x 2 4 x 2

2 x 1x 2 2

1 3x 2

2 x 1x 2 2

Fin

d th

e D

eriv

ativ

e

Find the derivative (slope of the tangent line) when x=1

1 3 1 2

2 1 1 1 2 2

28

14

Use the point-slope formula to find an equation

y 12 1

4 x 1

Example 3Differentiate the function:

1 21 12 3' d

dxf x x x

1 21 12 3

d ddx dxx x

2 31 12 31 2x x

2 31 2

2 3x x

Sum and Difference Rules

Constant Multiple Rule

Power Rule

Simplify

21 1

2 3x xf x

1 21 12 3

d ddx dxx x

The Quotient Rule is long, don’t forget to rewrite if possible.

Rewrite to use the power rule

33

xx 2

2

34 36

xx

3 3

3 46 6

xx x

Try to find the Least Common

Denominator

More Derivatives of Trigonometric Functions

We will assume the following to be true:

2tan secd

x xdx

sec sec tand

x x xdx

2cot cscd

x xdx

csc csc cotd

x x xdx

Example 1Differentiate the function:

f ' dd

sec1tan

1tan sec tan sec sec2

1tan 2

sec tan 1 1tan 2

Quotient Rule

f sec1tan

1tan dd sec sec d

d 1tan 1tan 2

u

v

u v'v u'

Use the Quotient Rule

sec tan sec tan 2 sec3 1tan 2

Always look to simplify

sec tan tan 2 sec2

1tan 2 Trig Law: 1 + tan2 = sec2

Example 2Differentiate the function:

f ' dd 3sec

3sectan sec3

3 sectan 3sec

Product Rule

f 3 sec

3 dd sec sec d

d 3

u v

u v' v u'

Use the Product Rule

Example 1Let

a. Find the derivative of the function.

f x 2x 4 9x 3 5x 2 7

ddx 2x 4 9x 3 5x 2 7

ddx 2x 4 d

dx 9x 3 ddx 5x 2 d

dx 7

2 ddx x 4 9 d

dx x 3 5 ddx x 2 d

dx 7

24x 4 1 93x 3 1 52x 2 1 0

8x 3 27x 2 10x

Example 1 (Continued)Let

b. Find the derivative of the function found in (a).

f x 2x 4 9x 3 5x 2 7

ddx 8x 3 27x 2 10x

ddx 8x 3 d

dx 27x 2 ddx 10x1

8 ddx x 3 27 d

dx x 2 10 ddx x1

83x 3 1 272x 2 1 101x1 1

24x 2 54x 10

Example 1 (Continued)Let

c. Find the derivative of the function found in (b).

f x 2x 4 9x 3 5x 2 7

ddx 24x 2 54x 10

ddx 24x 2 d

dx 54x1 ddx 10

24 ddx x 2 54 d

dx x1 ddx 10

242x 2 1 541x1 1 0

48x 54

Example 1 (Continued)Let

d. Find the derivative of the function found in (c).

f x 2x 4 9x 3 5x 2 7

ddx 48x 54

ddx 48x1 d

dx 54

48 ddx x1 d

dx 54

481x1 1 0

48

Example 1 (Continued)Let

e. Find the derivative of the function found in (d).

f x 2x 4 9x 3 5x 2 7

ddx 48

0

We have just differentiated the derivative of a function. Because the derivative of a function is a function,

differentiation can be applied over and over as long as the derivative is a differentiable function.

Higher-Order Derivatives: Notation

First Derivative:

Second Derivative:

Third Derivative:

Fourth Derivative:

nth Derivative:

y' f ' x dydx

ddx f x

2 2

2 2'' '' dy d yd ddx dx dx dx

y f x f x

y' ' ' f ' ' ' x ddx

d 2y

dx 2 d 3y

dx 3 d 3

dx 3 f x

y 4 f 4 x ddx

d 3y

dx 3 d 4 y

dx 4 d 4

dx 4 f x

y n f n x ddx

d n 1y

dx n 1 d ny

dx n d n

dx n f x Notice that for derivatives of higher order than the third, the

parentheses distinguish a derivative from a power. For example: .

f 4 x f 4 x

Example 1 (Continued)Let

f. Define the derivatives from (a-e) with the correct notation.

f x 2x 4 9x 3 5x 2 7

5 0f x

You should note that all higher-order derivatives of a

polynomial p(x) will also be polynomials, and if p has degree n, then p(n)(x) = 0 for

k ≥ n+1.

f 4 x 48

f ' ' ' x 48x 54

f ' ' x 24x 2 54 x 10

f ' x 8x 3 27x 2 10x

Example 2If , find . Will ever equal 0?

f x 1x

ddx x 1

1x 1 1

x 2

f 3 x

f n x Find the first derivative:

2ddx x

1 2x 2 1

2x 3

Find the second derivative:

32ddx x

2 3x 3 1

6x 4

Find the third derivative:

f ' ' ' x 6x 4

No higher-order derivative will equal 0 since the power of the function will never be 0. It decreases by one each time.

Example 3Find the second derivative of .

s t t 2 sin t 3t

s' t ddt t 2 sin t 3t

ddt t 2 sin t d

dt 3t

t 2 ddt sin t sin t d

dt t 2 ddt 3t

t 2cos t sin t2t 3

t 2 cos t 2t sin t 3

Find the first derivative:

s' ' t ddt t 2 cos t 2t sin t 3

ddt t 2 cos t d

dt 2t sin t ddt 3

t 2 ddt cos t cos t d

dt t 2 2t ddt sin t sin t d

dt 2t ddt 3

t 2 cos t cos t2t 2tcos t sin t2 0

t 2 cos t 2t cos2t cos t 2sin t

Find the second derivative:

Average Acceleration

Example: Estimate the velocity at time 5 for graph of velocity at time t below.

1 2 3 4 5 6

2

-2

-4

v(t)

t

Acceleration is the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.

Find the average rate of change of velocity for times that are close and enclose time 5.

6 4

6 4

v v 4 6

6 4

5

Instantaneous AccelerationIf s = s(t) is the position function of an object that moves in

a straight line, we know that its first derivative represents the velocity v(t) of the object as a function of time.

The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of an object. Thus, the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function.

a t v ' t s' ' t d 2sdt 2

v t s' t dsdt

Position, Velocity, and Acceleration

Position, Velocity, and Acceleration are related in the following manner:

Position:

Velocity:

Acceleration:

( )s t

'( ) ( )s t v t

Units = Measure of length (ft, m, km, etc)

The object is…Moving right/up when v(t) > 0Moving left/down when v(t) < 0Still or changing directions when v(t) = 0

Units = Distance/Time (mph, m/s, ft/hr, etc)Speed = absolute value of v(t)

s' ' t v ' t a t Units = (Distance/Time)/Time (m/s2)

Example

1 2 3 4 5 6

2

-2

-4

v(t)

t

Example: The graph below at left is a graph of a particle’s velocity at time t. Graph the object’s acceleration where it exists and answer the questions below

1 2 3 4 5 6

2

-2

-4

a(t)

t

m = 2

Cornerm = 0

m = -4

m = 4

When is the particle speeding up?

When is the particle traveling at a constant speed?

When is the function slowing down?

Positive acceleration and positive velocity

(0,2)

Negative acceleration and Negative velocity

U (5,6)

(2,4)

0 acceleration and 0 velocity

Negative acceleration and Positive velocity

(4,5) U (6,7)

Positive acceleration and Negative velocity

Moving away from x-axis.

ConstantMoving towards the x-axis.

Speeding Up and Slowing DownAn object is SPEEDING UP when the following occur:

• Algebraic: If the velocity and the acceleration agree in sign

• Graphical: If the curve is moving AWAY from the x-axis

An object is traveling at a CONSTANT SPEED when the following occur:

• Algebraic: Velocity is constant and acceleration is 0.

• Graphically: The velocity curve is horizontal

An object is SLOWING DOWN when the following occur:

• Algebraic: Velocity and acceleration disagree in sign

• Graphically: The velocity curve is moving towards the x-axis

Example 1The velocity of a particle moving left and right with respect to

an origin is graphed below. Complete the following:

1. Find the average acceleration between time 2 and time 6.

2. Graph the particle’s acceleration where it exists.

3. Describe the particles speed.

1 2 3 4 5 6

2

-2

-4

v(t)

t

Example 2

Sketch a graph of the function that describes the velocity of a particle moving up and down with the following characteristics:

The particle’s velocity is defined on [0,8]

The particle is only slowing down on

(1,3)U(4,7)

The acceleration is never 0.

Example 3The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(a) Find the acceleration at time t.

s t t 3 6t 2 9t

s' t v t ddt t 3 6t 2 9t

v t ddt t 3 d

dt 6t 2 ddt 9t

v t ddt t 3 6 d

dt t 2 9 ddt t1

v t 3t 3 1 62t 2 1 9t1 1

v t 3t 2 12t 9

The derivative

of the position

function is the

velocity function.

s' ' t v ' t a t ddt 3t 2 12t 9

a t ddt 3t 2 d

dt 12t ddt 9

a t 3 ddt t 2 12 d

dt t1 ddt 9

a t 32t 2 1 121t1 1 0

a t 6t 12

The derivative of the velocity function is

the acceleration

function.

Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(b) What is the acceleration after 4 seconds?

(c) Is the particle speeding up, slowing down, or traveling at a constant speed at 4 seconds?

s t t 3 6t 2 9t

a 4

6 4 12 12 m/s2

v 4

3 4 2 12 4 9

9 m/s

a 4 12 m/s2

Since the velocity and acceleration agree in signs, the particle is

speeding up.

Example 3 (continued)The position of a particle is given by the equation

where t is measured in seconds and s in meters.

(d) When is the particle speeding up? When is it slowing down?

s t t 3 6t 2 9t

Velocity:

Acceleration:

+ +–

+–

Speeding Up:

(1,2) U (3,∞)

Slowing Down:

(0,1) U (2,3)