Differentiating Quotients

The quotient rule tells us how to differentiate functions of the form $\frac{y}{z}.$ More specifically, we will verify the

Quotient Rule

$(\frac{y}{z})^\prime = \frac{y^\prime z - yz^\prime}{z^2}.$

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 $ x $ for which $z(x) \neq 0.$

To see why the Quotient Rule works the way it does, let's introduce the notation $w = \frac{y}{z}.$ This means that $ y = wz $ and we are tryinng to compute $w^\prime.$
Applying the Product Rule, we see that
$y^\prime = w^\prime z + wz^\prime.$

Solving for $w^\prime,$ we have
$\begin{eqnarray*}w^\prime & = & \frac{y^\prime - w z^\prime}{z} \\ & = & \frac{y^\prime - \frac{y}{z} z^\prime}{z} \\ & = & \frac{y^\prime z - y z^\prime}{z^2}.\end{eqnarray*}$

Example:

Consider the function $y(x) = \frac{x}{x^2 + 1}.$ If we want to compute the derivative of $ y, $ we find that

$y^\prime = \frac{x^\prime(x^2 + 1) - x(x^2 + 1)^\prime}{(x^2 + 1)^2} = \frac{(x^2 + 1) - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}.$
Below, the graph of the function $y(x) = \frac{x}{x^2 + 1}$ is shown on the left while its derivative $\frac{1 - x^2}{(x^2 + 1)^2}$ 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 $ x^n $ where $ n
$ is a negative exponent.
To see this, suppose that $ n $ is a negative exponent and let $ |n| $ denote its absolute value. Then
$\begin{eqnarray*} \frac{d}{dx}x^n & = & \frac{d}{dx}\frac{1}{x^{|n|}} \\ \\ & = & \frac{\frac{d}{dx}(1) x^{|n|} - 1\frac{d}{dx}x^{|n|}}{x^{2|n|}} \\ \\ & = & \frac{-|n| x^{|n| - 1}}{x^{2|n|}} \\ \\ & = & -|n|\frac{1}{x^{|n| + 1}} \\ \\ & = & nx^{n-1}. \end{eqnarray*}$

This means that we have verified that the Power Rule holds for any integer exponent. That is,
$\frac{d}{dx} x^n = nx^{n-1} \mbox{ for any integer } n.$