Derivatives of Quotients
Example:
A = (s2 - s) / (4*s3 - 2*s + 1)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| A |
= |
s2 - s |
 |
| 4*s3 - 2*s + 1 |
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,
| A |
= |
s2 - s |
 |
| 4*s3 - 2*s + 1 |
we can think of as
| A |
= |
f(s) |
 |
| g(s) |
So the derivative is
| A' |
= ( |
f(s) |
)' |
 |
| g(s) |
| |
= |
f '(s) |
g(s) |
- |
f(s) |
g '(s) |
 |
| ( g(s) )2 |
| |
= |
( s2 - s )' |
( 4*s3 - 2*s + 1 ) |
- |
( s2 - s ) |
( 4*s3 - 2*s + 1 )' |
 |
| ( 4*s3 - 2*s + 1 )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
| A' |
= |
( 2*s - 1 ) |
( 4*s3 - 2*s + 1 ) |
- |
( s2 - s ) |
( 4*3*s2 - 2 + 0 ) |
 |
| ( 4*s3 - 2*s + 1 )2 |
| |
= |
(2*s - 1) (4*s3 - 2*s + 1) - (s2 - s) (12*s2 - 2) |
 |
| (4*s3 - 2*s + 1)2 |
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Sat Jan 24 20:35:00 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.