Derivatives of Quotients
Example:
f(q) = sin(2*q - 2) / (4 + ln(3*q))
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(q) |
= |
sin(2*q - 2) |
 |
| 4 + ln(3*q) |
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(q) |
= |
sin(2*q - 2) |
|
| 4 + ln(3*q) |
we can think of as
| f(q) |
= |
g(q) |
|
| h(q) |
So the derivative is
| f(q)' |
= ( |
g(q) |
)' |
|
| h(q) |
| |
= |
g '(q) |
h(q) |
- |
g(q) |
h '(q) |
|
| ( h(q) )2 |
| |
= |
( sin(2*q - 2) )' |
( 4 + ln(3*q) ) |
- |
( sin(2*q - 2) ) |
( 4 + ln(3*q) )' |
|
| ( 4 + ln(3*q) )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(q)' |
= |
( cos(2*q - 2) (2 - 0) ) |
( 4 + ln(3*q) ) |
- |
( sin(2*q - 2) ) |
( 0 + 3*(3*q)-1 ) |
|
| ( 4 + ln(3*q) )2 |
| |
= |
2*cos(2*q - 2) (4 + ln(3*q)) - 3*sin(2*q - 2) (3*q)-1 |
|
| (4 + ln(3*q))2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Tue Mar 10 20:05:54 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.