Derivatives of Quotients

Example:
y = sin(q) / (1 + cos(q))

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

y = sin(q)
----------
1 + cos(q)

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 = sin(q)
----------
1 + cos(q)
we can think of as
y = f(q)
----------
g(q)
So the derivative is
y' = ( f(q) )'
----------
g(q)
  = f '(q) g(q) - f(q) g '(q)
----------
( g(q) )2
  = ( sin(q) )' ( 1 + cos(q) ) - ( sin(q) ) ( 1 + cos(q) )'
----------
( 1 + cos(q) )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(q) )' = cos(q) (by the derivative rules for basic functions)
( 1 + cos(q) )' = 0 + (-1)*sin(q) (by the derivative rule for sums, derivative rule for constants, and the derivative rules for basic functions)
so the finished derivative is
y' = ( cos(q) ) ( 1 + cos(q) ) - ( sin(q) ) ( 0 + (-1)*sin(q) )
----------
( 1 + cos(q) )2
  = cos(q) (1 + cos(q)) + sin(q) sin(q)
----------
(1 + cos(q))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.