Differentiate the following functions with respect to x :

$y=\tan ^{-1}\left(\frac{x}{1+6 x^{2}}\right)$

Arranging the terms in equation

$y=\tan ^{-1}\left(\frac{3 x-2 x}{1+3 x \times 2 x}\right)$

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

$y=\tan ^{-1}(3 x)-\tan ^{-1}(2 x)$

Differentiating w.r.t $\mathrm{x}$ we get

$\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1}(3 x)-\tan ^{-1}(2 x)\right)$

$\frac{d y}{d x}=\frac{3}{1+(3 x)^{2}}-\frac{2}{1+(2 x)^{2}}$

$\frac{d y}{d x}=\frac{3}{1+9 x^{2}}-\frac{2}{1+4 x^{2}}$