Derivatives of Quotients
Example:
f(t) = (t4 - 1) / (t + 1)9
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(t) |
= |
t4 - 1 |
 |
| (t + 1)9 |
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) |
= |
t4 - 1 |
|
| (t + 1)9 |
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 |
| |
= |
( t4 - 1 )' |
( (t + 1)9 ) |
- |
( t4 - 1 ) |
( (t + 1)9 )' |
|
| ( (t + 1)9 )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)' |
= |
( 4*t3 - 0 ) |
( (t + 1)9 ) |
- |
( t4 - 1 ) |
( 9*(t + 1)8 (1 + 0) ) |
|
| ( (t + 1)9 )2 |
| |
= |
4*t3 (t + 1)9 - 9*(t4 - 1) (t + 1)8 |
|
| ((t + 1)9)2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Fri Aug 29 23:36:31 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.