$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\tan }^{-1}\left(\frac{\mathrm{x}-\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}{\mathrm{x}+\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}\right)\right]=$

$$\begin{gathered} \mathrm{y}={\tan }^{-1}\left[\frac{\mathrm{x}-\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}{\mathrm{x}+\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}\right] \\ \mathrm{Put} \mathrm{x}=\mathrm{a} \sin \mathrm{\theta } \\ ={\tan }^{-1}\left[\frac{\mathrm{a} \sin \mathrm{\theta }-\mathrm{a} \cos \mathrm{\theta }}{\mathrm{a} \sin \mathrm{\theta }+\mathrm{a} \cos \mathrm{\theta }}\right] \\ ={\tan }^{-1}\left[\frac{\tan \mathrm{\theta }-1}{\tan \mathrm{\theta }+1}\right] \\ ={\tan }^{-1}\left[\tan \left(\mathrm{\theta }-\frac{\mathrm{\pi }}{4}\right)\right] \\ =\mathrm{\theta }-\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \mathrm{y}={\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right) \raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right. \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\sqrt{1+{\left({\displaystyle \frac{\mathrm{x}}{\mathrm{a}}}\right)}^2}}\frac{1}{\mathrm{a}} \\ =\frac{1}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}} \end{gathered}$$