Derivatives of Quotients
Example:
f(s) = sin(s - 5) / (s4 + s3 + 3)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| f(s) |
= |
sin(s - 5) |
 |
| s4 + s3 + 3 |
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(s) |
= |
sin(s - 5) |
|
| s4 + s3 + 3 |
we can think of as
| f(s) |
= |
g(s) |
|
| h(s) |
So the derivative is
| f(s)' |
= ( |
g(s) |
)' |
|
| h(s) |
| |
= |
g '(s) |
h(s) |
- |
g(s) |
h '(s) |
|
| ( h(s) )2 |
| |
= |
( sin(s - 5) )' |
( s4 + s3 + 3 ) |
- |
( sin(s - 5) ) |
( s4 + s3 + 3 )' |
|
| ( s4 + s3 + 3 )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(s)' |
= |
( cos(s - 5) (1 - 0) ) |
( s4 + s3 + 3 ) |
- |
( sin(s - 5) ) |
( 4*s3 + 3*s2 + 0 ) |
|
| ( s4 + s3 + 3 )2 |
| |
= |
cos(s - 5) (s4 + s3 + 3) - sin(s - 5) (4*s3 + 3*s2) |
|
| (s4 + s3 + 3)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 Dec 30 12:52:53 2025
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.