The Product and Quotient Rules

The Product Rule

 Let f and g be differentiable functions.  Then           (f(x)g(x))' = f(x)g'(x) + f '(x)g(x)

Proof:

We have

d/dx (fg)

f(x+h) g(x+h)  -  f(x) g(x)
=  lim
Add and subtract f(x + h)g(x)

h

f(x+h) g(x+h)  -  f(x+h) g(x)  +  f(x+h) g(x)  -  f(x) g(x)
=  lim

h

f(x+h) g(x+h)   -  f(x+h) g(x)               f(x+h) g(x)  -  f(x) g(x)
=  lim                                                      +

h                                                          h

g(x+h) - g(x)                  f(x+h) - f(x)
=  lim
f(x+h)                             +                              g(x)

h                                 h

=  [lim f(x+h)] g'(x) + g(x) f '(x)

=  f(x)g'(x) + g(x)f'(x)

Example

Find

d
(2 - x2) (x4 - 5)
dx

Solution:

Here

f(x) = 2 - x2

and

g(x) = x4 - 5

The product rule gives

d/dx [f(x)g(x)] = (2 - x2)(4x3) + (-2x)(x4 - 5)

The Quotient Rule

Remember the poem

"lo d hi minus hi d lo square the bottom and away you go"

This poem is the mnemonic for the taking the derivative of a quotient.

 d     f             g f '  - f g'                                                             dx    g                  g2

Example:

Find y' if

2x - 1
y =
x + 1

Solution:

Here

f(x) = 2x - 1

and

g(x) = x + 1

The quotient rule gives

(x + 1) (2)  -  (2x - 1) (1)
y'  =
(x + 1)2

2x + 2 - 2x + 1
=
(x + 1)2

3
=
(x + 1)2

Exercise

Suppose that the cost of producing x snowboards per hour is given by

50x + 1000
C = 100x  +
x + 2

find the marginal cost when x  =  10