$y={\mathrm{Cos}}^{-1}\left(\frac{a^2-x^2}{a^2+x^2}\right)+{\mathrm{Sin}}^{-1}\left(\frac{2ax}{a^2+x^2}\right)\Rightarrow \frac{dy}{dx}=$

$\text{Hint : Put}x=a\tan \theta$

$y={\mathrm{Cos}}^{-1}\left(\frac{a^2-a^2{\tan }^2\theta }{a^2+a^2{\tan }^2\theta }\right)+{\mathrm{Sin}}^{-1}\left(\frac{2a^2\tan \theta }{a^2+a^2{\tan }^2\theta }\right)$

$y={\cos }^{-1}\left(\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }\right)+{\sin }^{-1}\left(\frac{2\tan \theta }{1+{\tan }^2\theta }\right)$

$y={\cos }^{-1}\left(\cos 2\theta \right)+{\sin }^{-1}\left(\sin 2\theta \right)=4\theta$

$\frac{dy}{dx}=\frac{d}{dx}\left(4{\tan }^{-1}\frac{x}{a}\right)=\frac{4}{1+{\left({\displaystyle \frac{x}{a}}\right)}^2}\times \frac{1}{a}=\frac{4a}{x^2+a^2}$