Differentiate `tan^(-1)((sqrt(1+x^2)-1)/x)` with respect to `tan^(-1)x ,x!=0.`

Explanation

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

### Step 1: Define the functions
Let:
- \( u = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \)
- \( v = \tan^{-1}(x) \)

### Step 2: Differentiate \( u \) and \( v \) with respect to \( x \)
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