Question

Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

Hint:

When differentiating a function \(u\) with respect to another function \(v\), check if one can be written directly as a function of the other. If \(u = f(v)\), then \(\frac{du}{dv} = f'(v)\), eliminating any need to compute cumbersome derivatives with respect to \(x\).

Answer 1