Differentiate `tan^(-1)((sqrt(1+x^2)-1)/x) wrt sin ^(-1)((2x)/(1+x^2))dotofx (-1,1)`

Explanation

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

### Step 1: Simplify \( u \)

Let \( u = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \).

To simplify this, we can use the substitution \( x = \tan(\theta) \). Then, we have:

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