Differentiate `tan^(-1)((sqrt(1+x^(2))-1)/(x))` w.r.t. `tan^(-1)x.`

Explanation

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

### Step 1: Define the Functions
Let:
- \( A = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \)
- \( B = \tan^{-1}(x) \)

We need to find \( \frac{dA}{dB} \).
...