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

Explanation

To differentiate \( y = \sin^{-1}\left(\frac{2x}{1+x^2}\right) \) with respect to \( z = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \), we will use the chain rule.

### Step 1: Differentiate \( y \) with respect to \( x \)

Let \( y = \sin^{-1}\left(\frac{2x}{1+x^2}\right) \).

Using the derivative of \( \sin^{-1}(u) \), which is \( \frac{1}{\sqrt{1-u^2}} \), we first find \( u = \frac{2x}{1+x^2} \).

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