Differentiate tan^-1 (√(1 + x^2) - 1)/x with respect to sin^-1 2x/(1 + x^2), if -1 < x < 1, x ≠ 0.

Question

Differentiate tan^-1\({\frac{\sqrt{1 + x^2} - 1} {x}}\) with respect to sin^-1 2x/(1 + x²), if -1 < x < 1, x ≠ 0.

Answer 1

Suppose u =  tan^-1\({\frac{\sqrt{1 + x^2} - 1} {x}}\) and v = sin^-1 2x/(1 + x²)

To differentiate u with respect to v to find du/dv

On substituting x = tanθ in u

Given as -1 < x < 1

x ∈ (-1,1)

Here, x = tanθ

On differentiate u with respect to x 

On substituting x = tanθ in v

On differentiate v with respect to x