Derivatives of Quotients
Example:
f(x) = (ln(x) + ln(5)) / (x5 + p)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(x) |
= |
ln(x) + ln(5) |
 |
| x5 + 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(x) |
= |
ln(x) + ln(5) |
|
| x5 + p |
we can think of as
| f(x) |
= |
g(x) |
|
| h(x) |
So the derivative is
| f(x)' |
= ( |
g(x) |
)' |
|
| h(x) |
| |
= |
g '(x) |
h(x) |
- |
g(x) |
h '(x) |
|
| ( h(x) )2 |
| |
= |
( ln(x) + ln(5) )' |
( x5 + p ) |
- |
( ln(x) + ln(5) ) |
( x5 + p )' |
|
| ( x5 + 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(x)' |
= |
( x-1 + 0 ) |
( x5 + p ) |
- |
( ln(x) + ln(5) ) |
( 5*x4 + 0 ) |
|
| ( x5 + p )2 |
| |
= |
x-1 (x5 + p) - 5*x4 (ln(x) + ln(5)) |
|
| (x5 + 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 Jan 26 04:13:44 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.