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

If $u=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ and $v=\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$,then $\frac{d u}{d v}$ at $x=0$ is

$\lim _{x}$ ${\rightarrow \frac{\pi}{2}} \left( \frac{\int_{x^3}^{(\pi / 2)^3} (\sin (2 t^{1 / 3}) + \cos (t^{1 / 3})) dt}{(x - \frac{\pi}{2})^2} \right)$ is equal to:

$\frac{d}{dx} \left( \tan^{-1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) \right)$ is equal to

For $-1 < x < 1$,if $f(x) = \cos^2 \left( \tan^{-1} \sqrt{\frac{1-x}{1+x}} \right)$,then $f'(x) =$

If $f(x) = \sin^{-1}\left(\frac{2 \cdot 3^x}{1+9^x}\right)$,then $f^{\prime}\left(\frac{1}{2}\right)$ equals