Please, see my attached file and help me on the way solving a plate problem...
It is straightforward, you have a summation $\displaystyle u(x,y) = \sum_{n=1}^{\infty} F_n(x)G_n(y)$ where $\displaystyle F_n(x)$ is a function of $\displaystyle x$ only and $\displaystyle G_n(y)$ is a function of $\displaystyle y$ only then.
Then,
$\displaystyle \frac{\partial u}{\partial x} = \sum_{n=1}^{\infty}F_n'(x)G_(y)$
And,
$\displaystyle \frac{\partial u}{\partial y} = \sum_{n=1}^{\infty}F_n(x)G_n'(y)$.
And so on.