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

Question

Differentiate sin^-1\(\Big(\cfrac{2\text x}{1+\text x^2}\Big)\) with respect to tan^-1\(\Big(\cfrac{2\text x}{1-\text x^2}\Big)\), if -1 < x < 1.

Answer 1

Let u =  sin^-1\(\Big(\cfrac{2\text x}{1+\text x^2}\Big)\) and v =  tan^-1\(\Big(\cfrac{2\text x}{1-\text x^2}\Big)\).

We need to differentiate u with respect to v that is find \(\cfrac{du}{dv}. \)

We have u =  sin^-1\(\Big(\cfrac{2\text x}{1+\text x^2}\Big)\) 

By substituting x = tan θ, we have

On differentiating u with respect to x, we get

We have

We have