Differentiate `tan^(-1)((2x)/(1-x^2))` with respect to `sin^(-1)((2x)/(1+x^2))` , if `x in (-1,\ 1)`

Explanation

To differentiate \( \tan^{-1}\left(\frac{2x}{1-x^2}\right) \) with respect to \( \sin^{-1}\left(\frac{2x}{1+x^2}\right) \), we will follow these steps:

### Step 1: Define the functions
Let:
- \( y_1 = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \)
- \( y_2 = \sin^{-1}\left(\frac{2x}{1+x^2}\right) \)

We need to find \( \frac{dy_1}{dy_2} \).
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