PRODUCT, QUOTIENT, CHAIN RULES; DERIVATIES OF TRIG FUNCTIONS
The Product Rule
Now that you can find the derivatives of simple polynomials, it’s time to get more complicated.
What if you had to find the derivative of this?


f  x   x3  5 x 2  4 x  1 x5  7 x 4  x

You could multiply out the expression and take the derivative of each term. Let’s do this now:
Needless to say, this process is messy. Naturally, there’s an easier way. When a function
involves two terms multiplied by each other, we use the Product Rule, which is given below.
If f  x   uv , then f   x   u
dv
du
v
dx
dx
Let’s use the product rule to find the derivative from the example above:
Example 1

Example 2

f  x   9 x 2  4 x x3  7 x 2

Example 3
 1 1 1  1 1 1 
y    2  3   3  5 
x  x x
x 
x x
y

x  43 x
 x
5
 11x8 
The Quotient Rule
When you have to take the derivative of a function that is the quotient of two other functions,
you must use the quotient rule, given below.
du
dv
u
u
If f  x   , then f   x   dx 2 dx .
v
v
v
In this rule, as opposed to the Product Rule, the order in which you take the derivates is
important because of subtraction in the numerator. It’s always the bottom function times the
derivative of the top, minus the top function times the derivative of the bottom. Then divide the
whole thing by the bottom function squared. A good way to remember this is to say the
following:
LoDeHi  HiDeLo
 Lo 
Example 4
Example 5
 x  3x 
f  x 
 x  7x
5
2
Example 6
y
3x  5
5x  3
2
4
x
f  x 
x
3
 x 8
2
6
x


The Chain Rule
The most important rule (and sometimes the most difficult one) is called the Chain Rule. It’s
used when you’re given composite functions – that is, a function inside of another function.
A composite function is usually written as f  g  x   .
For example, if f  x  
1
1
and g  x   3x , then f  g  x   
.
x
3x
When finding the derivative of a composite function, we take the derivative the “outside”
function, with the “inside” function considered as the variable, leaving the “inside” function
alone. Then we multiply this result by the derivative of the “inside” function, with respect to its
variable.
Using notation, the chain rule is given below.
If y  f  u  and u  g  x  , then y  f  g  x   and
d
 f  g  x    f   g  x   g   x  .

dx 
Also, the chain rule may also be written as follows:
If y  f  u  and u  g  x  , then y  f  g  x   and
dy dy du

.
dx du dx
This rule is tricky, so here are several examples.
Example 7
Example 8

f  x   5 x3  3x

5
y  x3  4 x
Example 9
y

Example 10

x5  8 x3 x 2  6 x
 2x  8 
y 2

 x  10 x 

Example 11
y  8v 2  6v and v  5 x3  11x , then
5
Example 12
dy
?
dx
Find



2
dy
at x  1 if y   x3  x x 4  x 2 
dx
The Derivative of Trig Functions
Let’s find the derivatives for sine and cosine:
If y  sin x , then
dy
 cos x
dx
If y  cos x , then
dy
  sin x
dx
You need to know that in calculus, angles are always referred to as radians.
You should know the derivatives of all six trig functions. The good news is that these derivatives
are pretty easy, and all you have to do is memorize them.
Let’s find the derivative of the other four trig functions.
Example 13
If y  tan x 
Example 14
sin x
dy
, find
.
cos x
dx
Example 15
If y  sec x 
If y  cot x 
cos x
dy
, find
.
sin x
dx
Example 16
1
dy
, find
.
cos x
dx
If y  csc x 
1
dy
, find
.
sin x
dx
Summary:
Function
y  sin x
y  cos x
Derivative
y  cos x
y   sin x
y  tan x
y  sec2 x
y   csc2 x
y  sec x tan x
y   csc x cot x
y  cot x
y  sec x
y  csc x
Example 17
Example 18
Find the derivative of y  sin  5x 
Find the derivative of y  sec x 2
Example 19
Example 20
Find the derivative of y  sin  2 x 3 
Find the derivative of y  cos 3x
Example 21
Example 22
 x 
Find the derivative of y  tan 

 x 1 
Find the derivative of y  csc  x3  x  1
Find the derivative of each expression. Simplify when possible.
1.
4 x3  3x 2
5x7  1
2.
x
3.
 x  1
2
 4 x  3  x  1
10
x
4
 4x
2

4.
8
5.
 x 
 2 
 x 1
6.
4
7.
4 x8  x
8x4
8.
1  2 1 

 x   x  2 
x 
x 

9.
 x 


 x 1
10.
x
11.
12.
13.
4
100
x2  1
x2  1
 x  4  x  8 
 x  6  x  6 
x 6  4 x3  6
4
16.
x
17.
x2  2 x
x 4  x3
18.
 x
x
15.
x2  3
x 3
4
2
 x 2  2 x 3  x 
at x = 1
at x = 1
at x = 2
3
 2x  5 


 5x  2 
2
14.
x x


x x 
 2
2
19.
x4  x2
x
1  x 
2 2
at x = 1
ANSWERS
1.
80 x9  75 x8  12 x 2  6 x
5x
 1
7
2
14.
0
x 2  6 x  33
2.
3x 2  6 x  1
15.
3.
10  x  1
16.
6
17.

4.
5.
9
16 x 2  32
x2  4
3x 2  3x 4
x
2
 1
18.
4
3/ 4
6.
1  2x  5 


4  5x  2 
7.
32 x11  7 x 7 / 2
16 x8
8.
3x 2  1 
9.
10.
11.
 29 


  5 x  2 2 


1
3
 4
2
x
x
4 x3
 x  1
5
100  x 2  x 
99
x
2
12.
9
64
13.
106
 1
2 x
1/ 2
x
2
 2 x  1
 1
3/ 2
19.
 x  3
7
4
2 x2  1
x2  1

1
4
2
Find each.
1.
dy
1
if y  u 2  1 and u 
x 1
dx
2.
dy
t2  2
at x  1 if y  2
and t  x 3
dx
t 2
3.
dy
if y   x 6  6 x5  5 x 2  x  and x  t
dt
4.
du
1
at v  2 if if u  x3  x2 and x 
dv
v
5.
dy
1 u
at x  1 if y 
and u  x 2  1
1 u2
dx
6.
du
x
if u  y 3 and y 
and x  v 2
dv
x8
ANSWERS
2
1.

2.
24
3.
t
3
4.

7 2
16 3
5.
2
6.
 x  1
3
1

 6t 5/ 2   5 
2 t

48v5
v
2
 8
4



2
3/ 2
  5t  t  3t  15t 

Find the indicated derivative of each function.
1.
dy
Find
if y  sin 2 x
dx
15.
dy
2

Find
if y   1  cot 
dx
x

2.
Find
dy
if y  cos x 2
dx
16.
Find
3.
Find
dy
if y   tan x  sec x 
dx
4.
Find
dy
if y  cot 4 x
dx
5.
Find
dy
if y  sin 3 x
dx
6.
Find
dy
1  sin x
if y 
1  sin x
dx
7.
Find
dy
if y  csc2 x 2
dx
8.
Find
dy
if y  2sin 3 x cos 4 x
dx
9.
d4y
Find
if y  sin 2 x
dx 4
10.
Find
11.
d
 tan x 
 f  x   if f  x   
Find

dx
 1  tan x 
12.
Find
dr
if r    sec tan 2
d
13.
Find
dr
if r    cos 1  sin  
d
14.
Find
dr
sec 
if r   
d
1  tan 
dy
y  sin t  cos t and t  1  cos 2 x
dx
2

2
dy
if y  sin cos x
dx

Answers
1.
2sin x cos x
2.
2 x sin x 2
3.
2sec3 x  sec x
4.
4csc2  4x 
5.
3cos 3 x
2 sin 3 x
6.
(or sin 2x )
16.
2 cos 3 x
1  sin x 
2
7.
4 x csc 2  x 2  cot  x 2 
8.
6cos3x cos 4x  8sin3x sin 4x
9.
16sin 2x
10.
cos 1  cos 2 x   sin 1  cos 2 x   2sin x cos x 


11.
2 tan x sec 2 x
1  tan x 
3
12.
 sec   sec2  2    2    tan 2  sec  tan  
13.
 sin 1  sin     cos  
14.
15.
sec   tan   1
1  tan  

2
 4  2  2 
 2   csc   
 x 
 x 

 2 
1  cot  x  
 

3



cos cos x   sin x  1 




2 x