Product and Quotient rules
October 3, 2013
Product and Quotient rules
Product rule
Consider two functions f (x) = x 3 and g (x) = x 2 . Compare
d
(f (x) · g (x))
dx
and
d
d
(f (x)) ·
(g (x)).
dx
dx
Product and Quotient rules
Product rule
Theorem
0
f (x)g (x) = f 0 (x)g (x) + f (x)g 0 (x)
Product and Quotient rules
Examples
Find the derivative of
x 3 e 3x
Product and Quotient rules
Examples
Find the derivative of
x 3 e 3x
t 4 ln(t + 5)
Product and Quotient rules
Examples
Find the derivative of
x 3 e 3x
t 4 ln(t + 5)
(3x 2 + 6x)4x
Product and Quotient rules
Examples
Find the derivative of
x 3 e 3x
t 4 ln(t + 5)
(3x 2 + 6x)4x
e 2t
√
t
Product and Quotient rules
Quotient Rule
Theorem
f (x)
g (x)
0
=
f 0 (x)g (x) − f (x)g 0 (x)
(g (x))2
Product and Quotient rules
Examples
Find the derivative of the following functions
5x 2 + 3
x3 + 4
Product and Quotient rules
Examples
Find the derivative of the following functions
5x 2 + 3
x3 + 4
1
1 + e −x
Product and Quotient rules
Examples
Find the derivative of the following functions
5x 2 + 3
x3 + 4
1
1 + e −x
ex
x3
Product and Quotient rules