The Product and Quotient rules
The Product Rule
The derivative of a product of functions is not the product of the derivatives.
Examples:
a)
b)
f (x)  2 x  3
g(x)  3
f (x)g(x)  6x  9
f (x)  x  1
f (x)g(x)  x
g'(x)  0
f '(x)g'(x)  0
( fg )' ( x )  6
g(x)  x
2
f '(x)  2
x
f '(x)  1
g'(x)  1
( fg )' ( x )  2 x  1
To differentiate a product, use the product rule: (fg)' = f 'g + fg'
or (first)'(second) + (first)(second)'.
Examples: Use the shift rule also when needed.
1.
h ( x )  ( 2 x  3 )( 4 x  5 )
Check
it : Multiplyin g ( 2 x  3 )( 4 x  5 ) we have
h( x)  8 x
2.
3.
h ' ( x )  2 ( 4 x  5 )  ( 2 x  3 ) 4  16 x  2
2
 2 x  15
2
g ( x)  x ( x  2)
h( x)  5x
4. f ( x ) 
x 1
3
h ' ( x )  16 x  2
3
h ' ( x )  5 x  1  5 x[
1
2
( x  4)
2
g ' ( x )  2 x ( x  2 )  x 3( x  2 )
5
x
5. f ( x )  16 ( x  8 ) 2 ( x  9 ) 5 ( x  1) 3
( x  1)
1 2
2
3
2
 2 x( x  2)  3 x ( x  2)
5x
] 5 x 1 
2
x 1
2
The Quotient Rule
The derivative of a quotient is not the quotient of the derivatives.
d
f (x) 
Example:
1
 x
1
f '(x)   x
2
1

x
x
2
but
dx
d
(1 )

(x)
0
1
0.
dx
To differentiate a quotient use the quotient rule:
or

 f 
f ' g  fg '
  
2
g
 g 

T ' B  TB '
T 
  
2
B
B
Examples: Also use the shift rule where needed.
1.
h( x) 
x
2
 5x
h'( x) 
4x  3
( 2 x  5 )( 4 x  3 )  ( x
( 4 x  3)
2
 5 x )4
2
Never sqauare out the denominator, leave it as it is. Only simplify the numerator.
2. f ( x ) 
( x  3)
x
3. g ( x ) 
2
2
1
x
( x  1)
6
1 3
5
4. This example shows some quotients can be written as products and then differentiated
using the product rule.
g ( x) 
x
( x  1)
3