Derivatives of Quotients

Example:
R = (tan(s) + 5) / (cos(7*s) + 5)

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

R = tan(s) + 5
----------
cos(7*s) + 5

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,

R = tan(s) + 5
----------
cos(7*s) + 5
we can think of as
R = f(s)
----------
g(s)
So the derivative is
R' = ( f(s) )'
----------
g(s)
  = f '(s) g(s) - f(s) g '(s)
----------
( g(s) )2
  = ( tan(s) + 5 )' ( cos(7*s) + 5 ) - ( tan(s) + 5 ) ( cos(7*s) + 5 )'
----------
( cos(7*s) + 5 )2
and we just need to know each of the derivatives on the right-hand side of the equation. In this case these are
( tan(s) + 5 )' = 1 / (cos(s))2 + 0 (by the derivative rule for sums, derivative rules for basic functions, and the derivative rule for constants)
( cos(7*s) + 5 )' = 7*(-1)*sin(7*s) + 0 (by the derivative rule for sums, chain rule, and the derivative rule for constants)
so the finished derivative is
R' = ( 1 / (cos(s))2 + 0 ) ( cos(7*s) + 5 ) - ( tan(s) + 5 ) ( 7*(-1)*sin(7*s) + 0 )
----------
( cos(7*s) + 5 )2
  = (1 / (cos(s))2) (cos(7*s) + 5) + 7*(tan(s) + 5) sin(7*s)
----------
(cos(7*s) + 5)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.