2. It is straightforward, you have a summation $u(x,y) = \sum_{n=1}^{\infty} F_n(x)G_n(y)$ where $F_n(x)$ is a function of $x$ only and $G_n(y)$ is a function of $y$ only then.
$\frac{\partial u}{\partial x} = \sum_{n=1}^{\infty}F_n'(x)G_(y)$
$\frac{\partial u}{\partial y} = \sum_{n=1}^{\infty}F_n(x)G_n'(y)$.