Derivatives of Quotients

Example:
y = (12*x2 + 10) / (x4 + 10*x + 8)

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

y = 12*x2 + 10
----------
x4 + 10*x + 8

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,

y = 12*x2 + 10
----------
x4 + 10*x + 8
we can think of as
y = f(x)
----------
g(x)
So the derivative is
y' = ( f(x) )'
----------
g(x)
  = f '(x) g(x) - f(x) g '(x)
----------
( g(x) )2
  = ( 12*x2 + 10 )' ( x4 + 10*x + 8 ) - ( 12*x2 + 10 ) ( x4 + 10*x + 8 )'
----------
( x4 + 10*x + 8 )2
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( 12*x2 + 10 )' = 12*2*x + 0 (by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)
( x4 + 10*x + 8 )' = 4*x3 + 10 + 0 (by the derivative rule for sums, power rule, rule for constant multiples, and the derivative rule for constants)
so the finished derivative is
y' = ( 12*2*x + 0 ) ( x4 + 10*x + 8 ) - ( 12*x2 + 10 ) ( 4*x3 + 10 + 0 )
----------
( x4 + 10*x + 8 )2
  = 24*x (x4 + 10*x + 8) - (12*x2 + 10) (4*x3 + 10)
----------
(x4 + 10*x + 8)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.