The derivative of `tan^(-1)((sqrt(1+x^2)-1)/x)` with respect to `tan^(-1)((2xsqrt(1-x^2))/(1-2x^2))` at `x=0` is `1/8` (b) `1/4` (c) `1/2` (d) 1

Options

A. `1//8`
B. `1//4`
C. `1//2`
D. 1

Explanation

To solve the problem, we need to find the derivative of \( y = \tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) \) with respect to \( z = \tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right) \) at \( x = 0 \).

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