Differentiate tan^-1(√1 + x^2/x) w.r.t. tan^-1(2x√1 - x^2/1 - 2x^2)

Question

Differentiate \(tan^{-1}\left(\frac{\sqrt{1 + x^2}-1}{x}\right)\) w.r.t. \(tan^{-1}\left(\frac{2x\sqrt{1 - x^2}}{1 - 2x^2}\right).\)

tan^-1(√1 + x²/x)

tan^-1(2x√1 - x²/1 - 2x²)

Answer 1

tan^-1(√1 + x²/x)

Then we want to find du/dv

Put x = tan θ.

Then θ = tan^-1x

and