Find the second order derivatives of the function.

Let $y=\tan ^{-1} x$

Then,

$\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^{2}}$

$\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{1}{1+x^{2}}\right)=\frac{d}{d x}\left(1+x^{2}\right)^{-1}=(-1) \cdot\left(1+x^{2}\right)^{-2} \cdot \frac{d}{d x}\left(1+x^{2}\right)$

$=\frac{-1}{\left(1+x^{2}\right)^{2}} \times 2 x=\frac{-2 x}{\left(1+x^{2}\right)^{2}}$