Differentiate `sin^(-1)(2xsqrt(1-x^2))` with respect to `sec^(-1)(1/(sqrt(1-x^2)))` , if `x in (1//sqrt(2),\ 1)`

Explanation

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

### Step 1: Set up the functions
Let:
- \( u = \sin^{-1}(2x\sqrt{1-x^2}) \)
- \( v = \sec^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right) \)

We need to find \( \frac{du}{dv} \).
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