Section 3.3
The Product and Quotient Rules
and Higher-Order Derivatives
The Product Rule
d
[ f ( x) g ( x)]  f ( x) g ' ( x)  g ( x) f ' ( x)
dx
Note: The derivative of a product is NOT the
product of the derivatives. NO PARTIAL CREDIT
for committing this no-no.
Examples:
s 4  s

• #4
g ( s) 
• #14
f ( x)  ( x  2 x  1)( x  1)
• C=1
2
2
3
The Quotient Rule
d  f ( x)  g ( x) f ' ( x)  f ( x) g ' ( x)



g ( x ) 
dx  g ( x) 
2
• Aka lo ‘d hi minus hi ‘d lo over lolo
• Note: The derivative of a quotient is NOT the
quotient of the derivatives and NO PARTIAL
credit will be awarded for this no-no.
Examples:
• #8
• #44
t 2
g (t ) 
2t  7
2
sin x
f ( x) 
x
Trigonometric Derivatives
• From the quotient rule we get: MEMORIZE
THEM
d
tan x  sec x
dx
2
d
sec x  sec x tan x
dx
d
cot x   csc x
dx
2
d
csc x   csc x cot x
dx
Examples:
• #53
• #72
•
f ( x)  x tan x
2
x
e
f ( x) 
x4
(0, ¼)
Higher Order Derivatives
• Second Derivative Notations
2
y ' ' f ' ' ( x)
d y
dx
2
2
d
 f ( x)
dx
2
• For higher than second, you will have analogous
notations. See the listing on page 146 of your text.
•
More examples
• #78 Determine the points at which the graph
of the function has a horizontal tangent line.
2
x
f ( x) 
x 1
2
• #88, 98, 108