Use derivatives to sketch the graph of a function

Let

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

Find all points such that $f'(x) = 0$ ;
Determine the intervals on which $f$ is monotonic by examining the sign of $f'$ ;
Determine the intervals on which $f'$ is monotonic by examining the sign of $f''$ ;
Sketch the graph of $f$ .

We take the derivative,
$f'(x) = \frac{1-x^2}{(1+x^2)^2}.$

Thus,

$f'(x) = 0 \quad \implies \quad \frac{1-x^2}{(1+x^2)^2} = 0 \quad \implies \quad x = \pm 1.$
$f$ is increasing if $|x| < 1$ and decreasing if $|x| > 1$ .
Taking the second derivative,
$f''(x) = \frac{2x(x^2 - 3)}{(1+x^2)^3}.$

Thus, $f'$ is increasing if $-\sqrt{3} < x < 0$ or $x > \sqrt{3}$ and $f'$ is decreasing if $x < - \sqrt{3}$ or $0 < x < \sqrt{3}$ .
We sketch the curve,