Derivatives of Quotients
Example:
P = ln(q - 9) / (q3 + e)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| P |
= |
ln(q - 9) |
 |
| q3 + e |
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,
| P |
= |
ln(q - 9) |
|
| q3 + e |
we can think of as
| P |
= |
f(q) |
|
| g(q) |
So the derivative is
| P' |
= ( |
f(q) |
)' |
|
| g(q) |
| |
= |
f '(q) |
g(q) |
- |
f(q) |
g '(q) |
|
| ( g(q) )2 |
| |
= |
( ln(q - 9) )' |
( q3 + e ) |
- |
( ln(q - 9) ) |
( q3 + e )' |
|
| ( q3 + e )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
| P' |
= |
( (q - 9)-1 (1 - 0) ) |
( q3 + e ) |
- |
( ln(q - 9) ) |
( 3*q2 + 0 ) |
|
| ( q3 + e )2 |
| |
= |
(q - 9)-1 (q3 + e) - 3*q2 ln(q - 9) |
|
| (q3 + e)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Thu Feb 12 07:49:42 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.