Derivatives of Quotients

Example:
S = (sin(x) + 7) / (9*x4 + 5*x + 7)

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

S = sin(x) + 7
----------
9*x4 + 5*x + 7

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,

S = sin(x) + 7
----------
9*x4 + 5*x + 7
we can think of as
S = f(x)
----------
g(x)
So the derivative is
S' = ( f(x) )'
----------
g(x)
  = f '(x) g(x) - f(x) g '(x)
----------
( g(x) )2
  = ( sin(x) + 7 )' ( 9*x4 + 5*x + 7 ) - ( sin(x) + 7 ) ( 9*x4 + 5*x + 7 )'
----------
( 9*x4 + 5*x + 7 )2
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( sin(x) + 7 )' = cos(x) + 0 (by the derivative rule for sums, derivative rules for basic functions, and the derivative rule for constants)
( 9*x4 + 5*x + 7 )' = 9*4*x3 + 5 + 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
S' = ( cos(x) + 0 ) ( 9*x4 + 5*x + 7 ) - ( sin(x) + 7 ) ( 9*4*x3 + 5 + 0 )
----------
( 9*x4 + 5*x + 7 )2
  = cos(x) (9*x4 + 5*x + 7) - (sin(x) + 7) (36*x3 + 5)
----------
(9*x4 + 5*x + 7)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.