Solution
= Solution
First take $F_y=xe^{\sin(xy)}\cos(xy)$, then differentiate with respect to $x$. The <product rule> and <chain rule> give
$$
\boxed{F_{xy}=e^{\sin(xy)}\left[\cos(xy)+xy\bigl(\cos^2(xy)-\sin(xy)\bigr)\right]}.
$$
The <function> is smooth, so the two mixed <partial derivatives> agree.