Differentiation of $y = \tan^{-1} \Bigl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\Bigr\}$ How do i differentiate the following: $$y = \tan^{-1} \biggl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\biggr\}$$
I know that $\text{derivative}$ of $\tan^{-1}{x}$ is $\frac{1}{1+x^{2}}$ but not sure as to how to do this.

First the idea is to observe as to what would you substitute for $x$ in order to remove the square root. After some manipulations you find that the correct substitution is $x^{2} = \sin{2\theta}$. Once you have done this then you have $$y = \tan^{-1}\biggl\{ \frac{(\cos\theta + \sin\theta) - (\cos\theta - \sin\theta)}{(\cos\theta + \sin\theta)+(\cos\theta - \sin\theta)}\biggr\} = \theta = \frac{1}{2}\sin^{-1}{x^2}$$
So if $y = \frac{1}{2}\sin^{-1}{x^2}$ then $$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\sqrt{1-x^{4}}} \cdot 2x = \frac{x}{\sqrt{1-x^{4}}}$$
