If $\mathrm{f}(x)=\frac{\sin ^2 x}{1+\cot x}+\frac{\cos ^2 x}{1+\tan x}$, then the value of $\mathrm{f}^{\prime}\left(\frac{\pi}{6}\right)$ is equal to

If $y=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), 0 \leqslant x<\frac{\pi}{2}$, then $y^{\prime}\left(\frac{\pi}{6}\right)=$

The volume of tetrahedron with co-terminus edges $\bar{a}, \bar{b}, \bar{c}$ is $\frac{64}{3}$ cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ is _________ cubic units.

In a common emitter transistor amplifier circuit, the input resistance is $1.8 \mathrm{k} \Omega$ and output is obtained across a load resistance of $9 \mathrm{k} \Omega$. The alternating current gain is 70 . Corresponding to an a.c. input voltage of 6 mV , the output voltage will be