
Differentiating Quotients
The quotient rule tells us how to differentiate functions of the
form
More specifically, we will verify the
Quotient Rule
Of course, the mathematical gods do not allow us to divide by
zero, so we must state that the Quotient Rule only holds for values of
for which
To see why the Quotient Rule works the way it does, let's
introduce the notation
This means
that
and we are tryinng to compute
Applying the Product Rule, we see that
Solving for
we have

Example:
Consider the function
If we want to compute the derivative of
we find that
Below, the graph of the function
is shown on the left while its derivative
is shown on the right.
Questions:
 Can you explain the relationship between these
two graphs?
 What does it mean when the derivative is negative and
what does it mean when it is positive?
 Are the points where the
derivative equals zero significant?


More on the Power Rule
The Quotient Rule also allows us to extend the Power Rule to
functions of the form
where
is a negative exponent.
To see this, suppose that
is a negative
exponent and let
denote its absolute value. Then
This means that we have verified that the Power Rule holds for any
integer exponent. That is,

