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

Explanation

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

### Step 1: Differentiate \( y \) with respect to \( x \)

We will use the chain rule and the derivative of the inverse tangent function. The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is given by:

\[
\frac{d}{du} \tan^{-1}(u) = \frac{1}{1+u^2}
...