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

For $$x>1$$, if $$(2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}$$, then $$\left(1+\log _e 2 x\right)^2 \frac{d y}{d x}$$ is equal to

If $$\tan y=\frac{x \sin \alpha}{1-x \cos \alpha}$$ and $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{m}}{x^2+2 \mathrm{n} x+1}$$, then $$\mathrm{m}^2+\mathrm{n}^2$$ is

The approximate value of $$\sin \left(60^{\circ} 0^{\prime} 10^{\prime \prime}\right)$$ is (given that $$\sqrt{3}=1.732,1^{\circ}=0.0175^{\circ}$$ )