Derivatives of Quotients
Example:
z = (4*y + 1/2) / ln(7*y + 8)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| z |
= |
4*y + 1/2 |
 |
| ln(7*y + 8) |
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 |
= |
4*y + 1/2 |
|
| ln(7*y + 8) |
we can think of as
| z |
= |
f(y) |
|
| g(y) |
So the derivative is
| z' |
= ( |
f(y) |
)' |
|
| g(y) |
| |
= |
f '(y) |
g(y) |
- |
f(y) |
g '(y) |
|
| ( g(y) )2 |
| |
= |
( 4*y + 1/2 )' |
( ln(7*y + 8) ) |
- |
( 4*y + 1/2 ) |
( ln(7*y + 8) )' |
|
| ( ln(7*y + 8) )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' |
= |
( 4 + 0 ) |
( ln(7*y + 8) ) |
- |
( 4*y + 1/2 ) |
( (7*y + 8)-1 (7 + 0) ) |
|
| ( ln(7*y + 8) )2 |
| |
= |
4*ln(7*y + 8) - 7*(4*y + 1/2) (7*y + 8)-1 |
|
| (ln(7*y + 8))2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
previous page
Page Generated: Thu Feb 19 04:47:45 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.