Differentiate the following functions with respect to x :

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

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

$y=\tan ^{-1} x+\tan ^{-1} a$

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

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{a}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}+0$

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}$