Derivatives of Quotients
Example:
z = (tan(x) - 1) / (x + 1)2
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| z |
= |
tan(x) - 1 |
 |
| (x + 1)2 |
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,
| z |
= |
tan(x) - 1 |
|
| (x + 1)2 |
we can think of as
| z |
= |
f(x) |
|
| g(x) |
So the derivative is
| z' |
= ( |
f(x) |
)' |
|
| g(x) |
| |
= |
f '(x) |
g(x) |
- |
f(x) |
g '(x) |
|
| ( g(x) )2 |
| |
= |
( tan(x) - 1 )' |
( (x + 1)2 ) |
- |
( tan(x) - 1 ) |
( (x + 1)2 )' |
|
| ( (x + 1)2 )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
| z' |
= |
( 1 / (cos(x))2 - 0 ) |
( (x + 1)2 ) |
- |
( tan(x) - 1 ) |
( 2*(x + 1) (1 + 0) ) |
|
| ( (x + 1)2 )2 |
| |
= |
(1 / (cos(x))2) (x + 1)2 - 2*(tan(x) - 1) (x + 1) |
|
| ((x + 1)2)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 16 05:01:11 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.