1.6

Copyright © 2014 Pearson Education, Inc.

Differentiation Techniques:The Product and Quotient Rules

OBJECTIVE• Differentiate using the Product and the Quotient Rules.

• Use the Quotient Rule to differentiate the average cost, revenue, and profit functions.

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THEOREM 5: The Product Rule

Let Then, F(x) f (x)g(x).

F (x) d

dxf (x)g(x)

F (x) f (x)d

dxg(x) g(x)

d

dxf (x)

1.6 Differentiation Techniques: The Product and Quotient Rules

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Example 1: Find d

dxx4 2x3 7 3x2 5x .

1.6 Differentiation Techniques: The Product and Quotient Rules

4 3 22 7 3 5d

x x x xdx

4 3 2 3 22 7 6 5 3 5 4 6x x x x x x x

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1.6 Differentiation Techniques: The Product and Quotient Rules

Quick Check 1

Use the Product Rule to differentiate each of the following functions. Do not simplify.

a.)

b.)

5(2 1)(3 2)y x x x

5( 1)( )y x x x

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1.6 Differentiation Techniques: The Product and Quotient Rules

Quick Check 1 Solution

a.)

Using the Product Rule:

We get:

b.)

Again, using the Product Rule, we get:

5 1 1 5 1 1 1(2 1)(3 0) (3 2)(5 2 0)y x x x x x x

d d df x g x f x g x g x f x

dx dx dx

5 43(2 1) (3 2)(10 1)y x x x x

1 11 11 1 55 2

1 1( 1) ( ) 0

5 2y x x x x x x

5

5 4

1 11 1

25y x x x

xx

5(2 1)(3 2)y x x x

51y x x x

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THEOREM 6: The Quotient Rule If then, Q(x)

N (x)

D(x),

Q (x) D(x) N (x) N (x) D (x)

D(x) 2

1.6 Differentiation Techniques: The Product and Quotient Rules

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Example 2: Differentiate f (x) x2 3x

x 1.

1.6 Differentiation Techniques: The Product and Quotient Rules

2

2

2 3( )

( 1)

x xf x

x

2 2

2

2 5 3 3( )

( 1)

x x x xf x

x

2

2

( 1)(2 3) ( 3 )(1)( )

( 1)

x x x xf x

x

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1.6 Differentiation Techniques: The Product and Quotient Rules

Quick Check 2

a.) Differentiate: . Simplify your result.

b.) Show that

2

1 3( )

2

xf x

x

2

1

1 1

d ax a b

dx bx bx

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1.6 Differentiation Techniques: The Product and Quotient Rules

Quick Check 2 Solution

a.) Using the Quotient Rule:

We get:

2

( ) [ ( )] ( ) [ ( )]( )

( ) [ ( )]

d dg x f x f x g xd f x dx dx

dx g x g x

2

2 2

( 2)(0 3) (1 3 )(2 0)( )

( 2)

x x xf x

x

2 2

2 2

3 6 2 6( )

( 2)

x x xf x

x

2

2 2

3 2 6( )

( 2)

x xf x

x

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1.6 Differentiation Techniques: The Product and Quotient Rules

Quick Check 2 Solution Concluded

b.) Using the Quotient Rule:

We know that:

2

( ) [ ( )] ( ) [ ( )]( )

( ) [ ( )]

d dg x f x f x g xd f x dx dx

dx g x g x

2

1 ( 1)( ) ( 1)( )

1 1

d ax bx a ax b

dx bx bx

2

( ) ( )

( 1)

abx a abx b

bx

2( 1)

abx a abx b

bx

2( 1)

a b

bx

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DEFINITION:

If C(x) is the cost of producing x items, then the

average cost of producing x items is

If R(x) is the revenue from the sale of x items, then the

average revenue from selling x items is

If P(x) is the profit from the sale of x items, then the

average profit from selling x items is

C(x)

x.

R(x)

x.

P(x)

x.

1.6 Differentiation Techniques: The Product and Quotient Rules

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Example 3: Paulsen’s Greenhouse finds that the cost, in dollars, of growing x hundred geraniums is given by If the revenue from the sale of x hundred geraniums is given by find each of the following.

a) The average cost, the average revenue, and the average profit when x hundred geraniums are grown and sold.b) The rate at which average profit is changing when 300 geraniums are being grown.

C(x) 200 100 x4 .

R(x) 120 90 x ,

1.6 Differentiation Techniques: The Product and Quotient Rules

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Example 3 (continued):a) We let AC, AR, and AP represent average cost,

average revenue, and average profit.

A P (x) P(x)

x

R(x) C(x)

x

120 90 x 200 100 x4

x

A P (x) 80 90 x 100 x4

x

1.6 Differentiation Techniques: The Product and Quotient Rules

( ) 120 90( )R

R x xA x

x x

4( ) 200 100( )C

C x xA x

x x

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Example 3 (continued):b) First we must find AP(x). Then we can substitute 3

(hundred) into AP(x).

1.6 Differentiation Techniques: The Product and Quotient Rules

1 1 1 1

2 4 2 4

2

45 25 80 90 100x x x x

x

1 3 1 1

2 4 2 4

2

1 190 100 80 90 100 1

2 4x x x x x

x

PA x

A P (x) 80 90 x 100 x4

x

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Example 3 (concluded):

Thus, at 300 geraniums, Paulsen’s average profit is increasing by about $11.20 per plant.

1.6 Differentiation Techniques: The Product and Quotient Rules

1 1

2 4

2

80 45 75( )P

x xA x

x

1 1

2 4

2

80 45(3) 75(3)(3)

3PA

(3) 11.196PA

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1.6 Differentiation Techniques: The Product and Quotient Rules

Section Summary

• The Product Rule is:

• The Quotient Rule is:

• Be careful to note the order in which you write out the factors when using the Quotient Rule. Because the Quotient Rule involves subtraction and division, the order in which you perform the operations is important.

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

f x g x f x g x g x f xdx dx dx

2

( ) [ ( )] ( ) [ ( )]( )

( ) [ ( )]

d dg x f x f x g xd f x dx dx

dx g x g x