Derivatives of Quotients
Example:
y = e2*x / (1 - e(-1/2)*x)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| y |
= |
e2*x |
 |
| 1 - e(-1/2)*x |
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,
| y |
= |
e2*x |
|
| 1 - e(-1/2)*x |
we can think of as
| y |
= |
f(x) |
|
| g(x) |
So the derivative is
| y' |
= ( |
f(x) |
)' |
|
| g(x) |
| |
= |
f '(x) |
g(x) |
- |
f(x) |
g '(x) |
|
| ( g(x) )2 |
| |
= |
( e2*x )' |
( 1 - e(-1/2)*x ) |
- |
( e2*x ) |
( 1 - e(-1/2)*x )' |
|
| ( 1 - e(-1/2)*x )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
| y' |
= |
( 2*e2*x ) |
( 1 - e(-1/2)*x ) |
- |
( e2*x ) |
( 0 - (-1/2)*e(-1/2)*x ) |
|
| ( 1 - e(-1/2)*x )2 |
| |
= |
2*e2*x (1 - e(-1/2)*x) - (1/2)*e2*x e(-1/2)*x |
|
| (1 - e(-1/2)*x)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Fri Feb 20 12:31:42 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.