Derivatives of Quotients
Example:
R = sin(q) / (ln(q) + ln(A))
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| R |
= |
sin(q) |
 |
| ln(q) + ln(A) |
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,
| R |
= |
sin(q) |
|
| ln(q) + ln(A) |
we can think of as
| R |
= |
f(q) |
|
| g(q) |
So the derivative is
| R' |
= ( |
f(q) |
)' |
|
| g(q) |
| |
= |
f '(q) |
g(q) |
- |
f(q) |
g '(q) |
|
| ( g(q) )2 |
| |
= |
( sin(q) )' |
( ln(q) + ln(A) ) |
- |
( sin(q) ) |
( ln(q) + ln(A) )' |
|
| ( ln(q) + ln(A) )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
| R' |
= |
( cos(q) ) |
( ln(q) + ln(A) ) |
- |
( sin(q) ) |
( q-1 + 0 ) |
|
| ( ln(q) + ln(A) )2 |
| |
= |
cos(q) (ln(q) + ln(A)) - q-1 sin(q) |
|
| (ln(q) + ln(A))2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Sun Feb 15 14:49:59 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.