Derivatives of Quotients

Example:
f(t) = sin(8*t + 4) / ((1)/(2))t

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

f(t) = sin(8*t + 4)
----------
((1)/(2))t

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,

f(t) = sin(8*t + 4)
----------
((1)/(2))t
we can think of as
f(t) = g(t)
----------
h(t)
So the derivative is
f(t)' = ( g(t) )'
----------
h(t)
  = g '(t) h(t) - g(t) h '(t)
----------
( h(t) )2
  = ( sin(8*t + 4) )' ( ((1)/(2))t ) - ( sin(8*t + 4) ) ( ((1)/(2))t )'
----------
( ((1)/(2))t )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(8*t + 4) )' = cos(8*t + 4) (8 + 0) (by the chain rule)
( ((1)/(2))t )' = (ln(((1)/(2))))*((1)/(2))t (by the derivative rules for basic functions)
so the finished derivative is
f(t)' = ( cos(8*t + 4) (8 + 0) ) ( ((1)/(2))t ) - ( sin(8*t + 4) ) ( (ln(((1)/(2))))*((1)/(2))t )
----------
( ((1)/(2))t )2
  = 8*cos(8*t + 4) ((1)/(2))t - (((ln(((1)/(2))))))*sin(8*t + 4) ((1)/(2))t
----------
(((1)/(2))t)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.