Differentiate sin^-1(2x/(1 + x^2)) w.r.t cos^-1(1 - x^2/(1 + x^2))

Question

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

sin^-1(2x/(1 + x²)) w.r.t cos^-1(1 - x²/(1 + x²))

Answer 1

u = sin^-1(2x/(1 + x²)) and v = cos^-1(1 - x²/(1 + x²))

Then we want to find du/dv

Alternative Method :

Differentiating both sides w.r.t. v, we get

du/dv = 1.