Derivatives of Quotients
Example:
f(t) = sin(8*t + 4) / ((1)/(2))t
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(t) |
= |
sin(8*t + 4) |
 |
| ((1)/(2))t |
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(t) |
= |
sin(8*t + 4) |
|
| ((1)/(2))t |
we can think of as
| f(t) |
= |
g(t) |
|
| h(t) |
So the derivative is
| f(t)' |
= ( |
g(t) |
)' |
|
| h(t) |
| |
= |
g '(t) |
h(t) |
- |
g(t) |
h '(t) |
|
| ( h(t) )2 |
| |
= |
( sin(8*t + 4) )' |
( ((1)/(2))t ) |
- |
( sin(8*t + 4) ) |
( ((1)/(2))t )' |
|
| ( ((1)/(2))t )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(t)' |
= |
( cos(8*t + 4) (8 + 0) ) |
( ((1)/(2))t ) |
- |
( sin(8*t + 4) ) |
( (ln(((1)/(2))))*((1)/(2))t ) |
|
| ( ((1)/(2))t )2 |
| |
= |
8*cos(8*t + 4) ((1)/(2))t - (((ln(((1)/(2))))))*sin(8*t + 4) ((1)/(2))t |
|
| (((1)/(2))t)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
previous page
Page Generated: Wed Mar 11 03:56:02 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.