Derivative of $\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$ with respect to $\tan ^{-1} x$ for $-1 < x < 1$ is:
- A$2$
- B$\frac{1}{1+x^2}$
- C$\frac{2}{1+x^2}$
- D$\frac{1}{2}$
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