Differentiate the following functions with respect to x :

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

Dividing numerator and denominator by $b$

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

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

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

Differentiating w.r.t $x$ we get

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

$\frac{d y}{d x}=0+\frac{1}{1+x^{2}}$

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