Derivatives of Quotients

Example:
z = (x2 - x - 8) / e3*x

The first thing to notice when finding the derivative of this function is that it is a quotient, as shown below:

z = x2 - x - 8
----------
e3*x

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,

z = x2 - x - 8
----------
e3*x
we can think of as
z = f(x)
----------
g(x)
So the derivative is
z' = ( f(x) )'
----------
g(x)
  = f '(x) g(x) - f(x) g '(x)
----------
( g(x) )2
  = ( x2 - x - 8 )' ( e3*x ) - ( x2 - x - 8 ) ( e3*x )'
----------
( e3*x )2
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( x2 - x - 8 )' = 2*x - 1 - 0 (by the derivative rule for sums, power rule, derivative rule for variables, and the derivative rule for constants)
( e3*x )' = 3*e3*x (by the chain rule)
so the finished derivative is
z' = ( 2*x - 1 - 0 ) ( e3*x ) - ( x2 - x - 8 ) ( 3*e3*x )
----------
( e3*x )2
  = (2*x - 1) e3*x - 3*(x2 - x - 8) e3*x
----------
(e3*x)2
[]


additional explanation for the quotient rule
see another quotient rule example
practice gateway test
previous page
Page Generated: Sun Jan 25 15:52:40 2026
Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.