Differentiate the following functions with respect to x :

$y=\tan ^{-1}\left(\frac{5 x}{1-6 x^{2}}\right)$

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 $x$ we get

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

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

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