Derivatives of Quotients
Example:
V = sqrt(10*p) / (p3 - 1/4)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| V |
= |
sqrt(10*p) |
 |
| p3 - 1/4 |
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,
| V |
= |
sqrt(10*p) |
|
| p3 - 1/4 |
we can think of as
| V |
= |
f(p) |
|
| g(p) |
So the derivative is
| V' |
= ( |
f(p) |
)' |
|
| g(p) |
| |
= |
f '(p) |
g(p) |
- |
f(p) |
g '(p) |
|
| ( g(p) )2 |
| |
= |
( sqrt(10*p) )' |
( p3 - 1/4 ) |
- |
( sqrt(10*p) ) |
( p3 - 1/4 )' |
|
| ( p3 - 1/4 )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
| V' |
= |
( 10*(1/2)*(10*p)-1/2 ) |
( p3 - 1/4 ) |
- |
( sqrt(10*p) ) |
( 3*p2 - 0 ) |
|
| ( p3 - 1/4 )2 |
| |
= |
5*(10*p)-1/2 (p3 - 1/4) - 3*p2 sqrt(10*p) |
|
| (p3 - 1/4)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Mon Feb 2 20:51:25 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.