Derivatives of Quotients

Example:
A = (s2 - s) / (4*s3 - 2*s + 1)

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

A = s2 - s
----------
4*s3 - 2*s + 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,

A = s2 - s
----------
4*s3 - 2*s + 1
we can think of as
A = f(s)
----------
g(s)
So the derivative is
A' = ( f(s) )'
----------
g(s)
  = f '(s) g(s) - f(s) g '(s)
----------
( g(s) )2
  = ( s2 - s )' ( 4*s3 - 2*s + 1 ) - ( s2 - s ) ( 4*s3 - 2*s + 1 )'
----------
( 4*s3 - 2*s + 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
( s2 - s )' = 2*s - 1 (by the derivative rule for sums, power rule, and the derivative rule for variables)
( 4*s3 - 2*s + 1 )' = 4*3*s2 - 2 + 0 (by the derivative rule for sums, rule for constant multiples, rule for constant multiples (again), and the derivative rule for constants)
so the finished derivative is
A' = ( 2*s - 1 ) ( 4*s3 - 2*s + 1 ) - ( s2 - s ) ( 4*3*s2 - 2 + 0 )
----------
( 4*s3 - 2*s + 1 )2
  = (2*s - 1) (4*s3 - 2*s + 1) - (s2 - s) (12*s2 - 2)
----------
(4*s3 - 2*s + 1)2
[]

additional explanation for the quotient rule
see another quotient rule example
practice gateway test
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Comments to Gavin LaRose
glarose@umich.edu
©2001 Gavin LaRose, University of Michigan Math Dept.