If $x \cdot \log _e\left(\log _e x\right)-x^2+y^2=4(y>0)$, then $\frac{d y}{d x}$ at $x=\mathrm{e}$ is

If $u=\log (\sqrt{x-1}-\sqrt{x+1})$ and $v=\sqrt{x+1}+\sqrt{x-1}$ then $\frac{d u}{d v}=\ldots$.

If $\mathrm{f}(x)=3 x^2+2 x \mathrm{f}^{\prime}(1)+\mathrm{f}^{\prime \prime}(2)$, then $\mathrm{f}(x)=\ldots \ldots .$.

If $\mathrm{f}(x)=\sqrt{1+\cos ^2\left(x^2\right)}$, then $\mathrm{f}^{\prime}\left(\frac{\sqrt{\pi}}{2}\right)$ is