The value of $\frac{d}{dx}(\log_e x)$ is:

$\frac{d}{dx} \sqrt{\sin 2x}$ is equal to

$\int \frac{dx}{{(2ax - x^2)}^{1/2}} = a^n \sin^{-1} \left( \frac{x}{a} - 1 \right)$. In this formula,$n =$ . . . . . . .

The value of $\frac{d}{dx} (\cos 4x^2)$ is .......

If $F = \frac{2}{\sin \theta + \sqrt{3} \cos \theta}$,then the minimum value of $F$ is

If $y = \frac{1}{a - z}$,then $\frac{dz}{dy} = $