Derivatives of Quotients
Example:
G = cos(8*x - 8) / (5*x3 - 5)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| G |
= |
cos(8*x - 8) |
 |
| 5*x3 - 5 |
The Derivative Rule for Quotients:
The derivative of a quotient is the derivative of the numerator
times the denominator minus the numerator times the derivative of the
denominator, all divided by the denominator squared.
If
| |
z |
= ( |
f(x) |
) |
|
| g(x) |
then the derivative of
z is
| |
z ' |
= ( |
f(x) |
)' |
|
| g(x) |
| |
|
= |
f '(x) g(x) |
- |
f(x) g '(x) |
|
|
|
( g(x) )2 |
So our example,
| G |
= |
cos(8*x - 8) |
|
| 5*x3 - 5 |
we can think of as
| G |
= |
f(x) |
|
| g(x) |
So the derivative is
| G' |
= ( |
f(x) |
)' |
|
| g(x) |
| |
= |
f '(x) |
g(x) |
- |
f(x) |
g '(x) |
|
| ( g(x) )2 |
| |
= |
( cos(8*x - 8) )' |
( 5*x3 - 5 ) |
- |
( cos(8*x - 8) ) |
( 5*x3 - 5 )' |
|
| ( 5*x3 - 5 )2 |
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
so the finished derivative is
| G' |
= |
( (-1)*sin(8*x - 8) (8 - 0) ) |
( 5*x3 - 5 ) |
- |
( cos(8*x - 8) ) |
( 5*3*x2 - 0 ) |
|
| ( 5*x3 - 5 )2 |
| |
= |
(-8)*sin(8*x - 8) (5*x3 - 5) - 15*x2 cos(8*x - 8) |
|
| (5*x3 - 5)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
previous page
Page Generated: Sat Apr 11 10:49:55 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.