Derivatives of Quotients
Example:
C = sqrt(8*x) / ln(x2 + 2)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| C |
= |
sqrt(8*x) |
 |
| ln(x2 + 2) |
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,
| C |
= |
sqrt(8*x) |
|
| ln(x2 + 2) |
we can think of as
| C |
= |
f(x) |
|
| g(x) |
So the derivative is
| C' |
= ( |
f(x) |
)' |
|
| g(x) |
| |
= |
f '(x) |
g(x) |
- |
f(x) |
g '(x) |
|
| ( g(x) )2 |
| |
= |
( sqrt(8*x) )' |
( ln(x2 + 2) ) |
- |
( sqrt(8*x) ) |
( ln(x2 + 2) )' |
|
| ( ln(x2 + 2) )2 |
and we just need to know each of the derivatives on the right-hand
side of the equation. In this case these are
| ( sqrt(8*x) )' |
= |
8*(1/2)*(8*x)-1/2 |
(by the chain rule) |
| ( ln(x2 + 2) )' |
= |
(x2 + 2)-1 (2*x + 0) |
(by the chain rule) |
so the finished derivative is
| C' |
= |
( 8*(1/2)*(8*x)-1/2 ) |
( ln(x2 + 2) ) |
- |
( sqrt(8*x) ) |
( (x2 + 2)-1 (2*x + 0) ) |
|
| ( ln(x2 + 2) )2 |
| |
= |
4*(8*x)-1/2 ln(x2 + 2) - 2*x sqrt(8*x) (x2 + 2)-1 |
|
| (ln(x2 + 2))2 |
![[]](/images/boxby.gif)
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Sat Mar 7 01:42:39 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.