Derivatives of Quotients
Example:
h(z) = (tan(z) - 1) / (z3 + 1)
The first thing to notice when finding the derivative of this function
is that it is a quotient, as shown below:
| h(z) |
= |
tan(z) - 1 |
 |
| z3 + 1 |
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,
| h(z) |
= |
tan(z) - 1 |
 |
| z3 + 1 |
we can think of as
| h(z) |
= |
f(z) |
 |
| g(z) |
So the derivative is
| h(z)' |
= ( |
f(z) |
)' |
 |
| g(z) |
| |
= |
f '(z) |
g(z) |
- |
f(z) |
g '(z) |
 |
| ( g(z) )2 |
| |
= |
( tan(z) - 1 )' |
( z3 + 1 ) |
- |
( tan(z) - 1 ) |
( z3 + 1 )' |
 |
| ( z3 + 1 )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
| h(z)' |
= |
( 1 / (cos(z))2 - 0 ) |
( z3 + 1 ) |
- |
( tan(z) - 1 ) |
( 3*z2 + 0 ) |
 |
| ( z3 + 1 )2 |
| |
= |
(1 / (cos(z))2) (z3 + 1) - 3*z2 (tan(z) - 1) |
 |
| (z3 + 1)2 |
additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Page Generated: Tue Feb 24 17:01:51 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose,
University of Michigan Math Dept.