Derivatives of Quotients
Example:
f(p) = (1 + 5*p) / tan(2*p)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(p) |
= |
1 + 5*p |
 |
| tan(2*p) |
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,
| f(p) |
= |
1 + 5*p |
|
| tan(2*p) |
we can think of as
| f(p) |
= |
g(p) |
|
| h(p) |
So the derivative is
| f(p)' |
= ( |
g(p) |
)' |
|
| h(p) |
| |
= |
g '(p) |
h(p) |
- |
g(p) |
h '(p) |
|
| ( h(p) )2 |
| |
= |
( 1 + 5*p )' |
( tan(2*p) ) |
- |
( 1 + 5*p ) |
( tan(2*p) )' |
|
| ( tan(2*p) )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
| f(p)' |
= |
( 0 + 5 ) |
( tan(2*p) ) |
- |
( 1 + 5*p ) |
( 2*1 / (cos(2*p))2 ) |
|
| ( tan(2*p) )2 |
| |
= |
5*tan(2*p) - 2*(1 + 5*p) (1 / (cos(2*p))2) |
|
| (tan(2*p))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 02:41:53 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.