If \(\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)\), then \(\mathrm{f}^{\prime}(\mathrm{e})\) is

\(f(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right) \)
\( \therefore f^{\prime}(x) =\frac{1}{\sqrt{1-\left(\frac{2 \log x}{1+(\log x)^2}\right)^2}}\)
\(=\frac{1+(\log x)^2}{\sqrt{1+(\log x)^4+2(\log x)^2-4(\log x)^2}} \) \( \times \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{2 \log x}{1+(\log x)^2}\right) \)
\( =\frac{1+(\log x)^2}{\sqrt{1-2(\log x)^2+(\log x)^4}} \) \( \times \frac{\left[1+(\log x)^2\right] \times \frac{2}{x}-(2 \log x)\left(\frac{2 \log x}{x}\right)}{\left[1+(\log x)^2\right]^2} \)
\( =\frac{1}{1-(\log x)^2} \times \frac{2+2(\log x)^2-4(\log x)^2}{x\left[1+(\log x)^2\right]} \)
\( =\frac{1}{1-(\log x)^2} \times \frac{2\left[1-(\log x)^2\right]}{x\left[1+(\log x)^2\right]} \)
\( =\frac{2}{x\left[1+(\log x)^2\right]} \)
\( \therefore \mathrm{f}^{\prime}(\mathrm{e})=\frac{1}{\mathrm{e}} \)