Product and
Quotient Rules
Other Trig Function Derivatives
Memorize
d
d
cos x    sin x
sin x   cos x

dx
dx
d
2
 tan x   sec x d cot x    csc2 x
dx
dx
d
sec x   sec x tan x
d
dx
csc x    csc x cot x
dx
Ex. 1 Find the derivative of y  x  tan x
y  1  sec x
2
1  cos x
y
sin x
Ex. 2 Find the derivative of
1
cos x

Separate: y 
sin x sin x
Rewrite:
y  csc x  cot x
y   csc x cot x  csc x
2
Product Rule (Memorize)
d
 f  x  g  x    f   x  g  x   g   x  f  x 
dx
• The derivative of the first times the second
plus the derivative of the second times the
first.
Ex. 3 Find the derivative of
h  x    3x  2 x
2
 5  4x 
Derivative of first
Derivative of
second
h  x    3  4 x  5  4 x    4   3x  2 x
h  x   24 x  4 x  15
2
2

Ex. 4 Find the derivative of
r  x   x sin x
Derivative of first
Derivative of
second
r   x   1 sin x   cos x  x 
r   x   sin x  x cos x
Ex. 5 Find the derivative of
Position
s  t   3t  t  5 
2
Derivative of first
Derivative of
second
s  t   v  t    6t  t  5   1  3t
Velocity
Acceleration
2

s  t   v  t   a  t    6  t  5   1 6t   6t
Ex. 6 Find the derivative of
h  x   x  x  5x 
3
h  x  x
1
2
Derivative of first
x
3
 5x 
Derivative of
second
1 1 2  3

2
h  x    x   x  5 x    3 x  5 
2

 x
Quotient Rule (Memorize)
d  f  x  f  x g  x  g x f  x


2
dx  g  x  
 g  x 
• The derivative of the top times the bottom
minus the derivative of the bottom times
the top all over the bottom squared.
Ex. 7 Find the derivative of
d  5x  2 
2

dx  x  1 
Derivative of top

 5  x
2
Derivative of
bottom
 1   2 x  5 x  2 
x
2
 1
2
Ex. 8 Find the derivative of
Rewrite
d  3  x 1 
dx  x  5 
Derivative of top
1

3 
d 
x


dx  x  5 


Derivative of
bottom
x   x  5   1  3  x 


2
1
 x  5
2
Ex. 9 Find the derivative of
1


2
d
x
sin
x
Rewrite
 2

dx   x  3 


Derivative of top
 
d  x sin x 
 2

dx   x  3 


Derivative of
bottom

1
1
 1 1 2
2
2 
2
x
sin
x

cos
x
x
x

3

2
x
x
sin x








2



2
2
 x  3

Suppose u and v are functions of x that are
differentiable at x=0 and that
u  0   5, u   0   3, v  0   1, and v  0   2
d
uv 
dx
  3 1   2  5 
 3  10
 13
d u 
dx  v 
3 1   2  5 


2
 1
3  10

1
 7