11.3:Derivatives of Products and Quotients

Can you find the derivatives of the following functions?

1) f(x) = x2 x4

2) f(x) = 3x3 (2x2 – 3x + 1)3) f(x) = 5x8 ex

4) f(x) = x7 ln x

** You can find the derivative for problem 1 and 2 easily by multiply two functions, but not for problem3 and 4.** You can find the derivative for allof the above problems using the limit definition, but it can be a long and tedious process

As you know in section 10.5, the derivative of a sum is the sum of the derivatives.

F(x) = u(x) + v(x)

F’(x) = u’(x) + v’(x)

Is the derivative of a product the product of the derivatives?

NO

Derivatives of Products

In words: The derivative of the 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.

Theorem 1 (Product Rule)

If f (x) = u(x) v(x), and if u ’(x) and v ’(x) exist, then

f ’ (x) = u(x) v ’(x) + u ’(x) v(x)

Or

f ’ (x) = u ’(x) v(x) + v ’(x) u(x)

Example 1) Find f’(x) if f(x) = x2 x4

Method 1

f(x) = x2 x4

f(x) = x6

f’(x) = 6x5

Method 2 – Apply product rule

F(x) = x2 x4

F’(x) = x2 (x4)’ + x4 (x2)’

F’(x) = x2 (4x3) + x4 (2x)

F’(x) = 4x5 + 2x5

F’(x) = 6x5

Example 2) Find f’(x) if f(x) = 3x3 (2x2 – 3x + 1)

Method 1f(x) = 3x3 (2x2 – 3x + 1)f(x) = 6x5 – 9x4 + 3x3

f’(x)= 30x4 – 36x3 + 9x2

Method 2: Apply product rulef(x) = 3x3 (2x2 – 3x + 1) f’(x)= 3x3 (2x2 – 3x + 1)’ + (2x2 – 3x + 1)(3x3)’ f’(x)= 3x3 (4x -3) + (2x2 – 3x + 1)(9x2) f’(x)= 12x4 – 9x3 +18x4 – 27x3 + 9x2

f’(x)= 30x4 – 36x3 + 9x2

Example 3) Find f’(x) if f(x) = 5x8 ex

f(x) = 5x8 ex

f’(x) = 5x8 (ex)’ + ex (5x8)’

f’(x) = 5x8 (ex) + ex (40x7)

f’(x) = 5x8 ex + 40 x7 ex

f’(x) = 5x7 ex (x + 8) or 5x7 (x + 8) ex

Note that the only way to do is to apply the product rule

Example 4) Find f’(x) if f(x) = x7 ln x

f(x) = x7 ln x

f’(x) = x7 (ln x)’ + ln x (x7)’

f’(x) = x7 (1/x) + ln x (7x6)

f’(x) = x6 + 7x6 ln x

f’(x) = x6 (1+ 7 lnx)

Example 5Let f(x) = (2x+9)(x2 -12)

A) Find the equation of the line tangent to the graph of f(x) at x = 3

f’(x) = (2x+9)(2x) + (x2 – 12)(2)

f’(x) = 4x2 + 18x + 2x2 – 24 = 6x2 + 18x - 24

so slope m = f’(3) = 6(3)2 + 18(3) – 24 = 84

also, if x = 3, f(3) = (2*3+9)(32 -12) = -45

y = mx + b

-45 = 84(3) + b

b = - 297

Therefore the equation of the line tangent is y = 84x - 297

B) Find the values(s) of x where the tangent line is horizontal

The slope of a horizontal line is 0 so set f’(x) = 0

6x2 + 18x – 24 = 0

6(x2 + 3x – 4) = 0

6(x + 4) (x – 1) = 0 so x = -4 or 1

Theorem 2 (Quotient Rule)

If f (x) = T (x) / B(x), and if T ’(x) and B ’(x) exist, then

Derivatives of Quotients

2)]([

)()(')()(')('

xB

xTxBxBxTxf

In words: The derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared.

Example 6

A) Find f’(x) for3

2)(

2

x

xxf

22

22

3

)2()'3()3()'2()('

x

xxxxxf

22

22

3

462)('

x

xxxf 22

2

3

26)('

x

xxf

22

2

3

)2)(2()3(2)('

x

xxxxf

Example 6 (continue)

B) Find f’(x) for4

3)(

2

3

t

ttxf

22

3223

)4(

)3()'4()4()'3()('

t

ttttttxf

22

24224

)4(

62123123)('

t

tttttxf

22

24

)4(

129)('

t

ttxf

22

322

)4(

)3)(2()4)(33()('

t

tttttxf

Example 6 (continue)

C) Find d/dx for

Method 1:

3

32)(

x

xxf

122

)(33

3

xx

xxf

12)( 3 xxf46)(' xxf

4

6)('

xxf

Method 2: apply quotient rule

3

32)(

x

xxf

23

3232

)(

)2)(3()(3)('

x

xxxxxf

6

525 363)('

x

xxxxf

46

2 66)('

xx

xxf

Example 7

A) Find f’(x) for2

)(3

xe

xxf

2

33

)2(

)'2()2()'()('

x

xx

e

xeexxf

2

322

)2(

63)('

x

xx

e

xexexxf

Note that the only way to do is to apply the quotient rule

2

32

)2(

)()2(3)('

x

xx

e

xeexxf

Example 7 (continue)

B) Find f’(x) forx

xxf

ln1

4)(

2ln1

)4)(/1()ln1(4)('

x

xxxxf

2ln1

4ln44)('

x

xxf

2ln1

ln4)('

x

xxf

2ln1

)4()'ln1()ln1()'4()('

x

xxxxxf

Example 8

The total sales S (in thousands of games) t months after the game

is introduced is given by

A) Find S’(t)

B) Find S(12) and S’(12). Explain

S(12) = 120 and S’(12) = 2

After 12 months, the total sale is120,000 games. Sales are

increasing at a rate of 2000 games per month.

C) Use the results above to estimate the total sales after 13 months

S(13) = S(12) + S’(12) = 122. The total sales after 13 months is

122,000 games.

3

150)(

t

ttS

2)3(

)150)(1()3(150

t

tt22 )3(

450

)3(

150450150

tt

tt