Question

Differentiate: \[ \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \quad \text{w.r.t.} \quad \cos^{-1}(2x\sqrt{1 - x^2}), \quad x \in \left(\frac{1}{\sqrt{2}}, 1\right) \]

Hint:

In problems involving inverse trigonometric functions, use substitution to simplify the differentiation, and don't forget to apply the product and chain rules.

Answer 1

Let \( y = \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \). The objective is to compute the derivative of \( y \) with respect to \( x \). This requires applying differentiation rules for inverse trigonometric functions and the product rule. The differentiation process is initiated as follows:\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\tan^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)}{x} \right)\]The subsequent steps involve algebraic simplification and the application of the chain rule to the inverse trigonometric component, leading to the final derivative.