The quotient rule

A function in the form of a fraction is differentiated with the quotient rule¹. We check it with $f(x) = \dfrac{x^2 - 1}{x^2 + 1}$ .

The formula

\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}

Taking $u = x^2 - 1$ and $v = x^2 + 1$ gives $u' = v' = 2x$ , so the derivative is as follows.

f'(x) = \frac{2x(x^2 + 1) - (x^2 - 1) \cdot 2x}{(x^2 + 1)^2} = \frac{4x}{(x^2 + 1)^2}

The numerator is $u'v - uv'$ , and reversing the order of the subtraction reverses the sign. Unlike the product rule, the order matters.

Monotonicity

The denominator is a square and so always positive; the sign comes from the numerator $4x$ alone.

Range	$f'$	Behavior
$x < 0$	negative	decreasing
$x > 0$	positive	increasing

At $x = 0$ the function attains its minimum $f(0) = -1$ .

Asymptote

As $x \to \pm\infty$ the numerator and the denominator have the same degree, so $f \to 1$ . The line $y = 1$ is an asymptote. Indeed the function can be rewritten as follows.

f(x) = 1 - \frac{2}{x^2 + 1}

It is $1$ minus a positive quantity. In this form the same answer follows from differentiating $-2(x^2+1)^{-1}$ , without the quotient rule at all.

The main values

$x$	$f(x)$	$f'(x)$
$-1$	$0$	$-1$
$0$	$-1$	$0$
$1$	$0$	$1$

Being even, the graph is symmetric about the $y$ -axis. The range is $-1 \leq f < 1$ , and the curve meets the $x$ -axis at the two points $x = \pm 1$ .

It follows from the product rule

Reading $\dfrac{u}{v}$ as $uv^{-1}$ and using the product rule with $\left( v^{-1} \right)' = -\dfrac{v'}{v^2}$ gives the following.

u'v^{-1} + u\left( -\frac{v'}{v^2} \right) = \frac{u'v - uv'}{v^2}

Anyone who would rather not memorize another formula can stop here.

Where the denominator vanishes the function is not defined in the first place, so it cannot be differentiated there either. For this function $x^2 + 1 > 0$ , so it is differentiable at every $x$ .

The upward-curving graph is $y = \dfrac{x^2 - 1}{x^2 + 1}$ , the curve through the origin is the derivative, and the large dots are the minimum and the crossings of the $x$ -axis.

Quotient rule, Wikipedia