## 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,