Differentiate the following functions with respect to x :

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

Arranging the terms in equation

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

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

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

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

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

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

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3}{1+\left(\mathrm{x}^{\frac{1}{3}}\right)^{2}} \times \frac{1}{3}\left(\mathrm{x}^{-\frac{2}{3}}\right)$

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