Solution
= Solution
The specified <unit vector> is $\mathbf u=(\cos(\pi/3),\sin(\pi/3))=(1/2,\sqrt3/2)$; the positive $y$ component selects that direction. The <gradient> is $\nabla f=e^{xy}(y,x)$, giving $(0,-2)$ at the point. The <directional derivative> is therefore
$$
\boxed{D_{\mathbf u}f=\nabla f\cdot\mathbf u=-\sqrt3}.
$$