If we're trying to extremize the functional, $I = \int\int f(u) dx dy$ , where: $$f(u) = u^2 + (\frac{\partial u}{\partial x})^2 + (\frac{\partial u}{\partial y})^2 + (\frac{\partial^2 u}{\partial x^2})^2 + (\frac{\partial^2 u}{\partial x \partial y})^2+ (\frac{\partial^2 u}{\partial y^2})^2$$
As far as my understanding goes, for a functional with higher order derivatives, then the E-L function would be something like: $$(\frac{\partial F}{\partial u}) - \frac{\mathrm{d}}{\mathrm{d}x}(\frac{\partial F}{\partial u_x}) - \frac{\mathrm{d}}{\mathrm{d}y}(\frac{\partial F}{\partial u_y})+ \frac{\mathrm{d^2}}{\mathrm{d}x^2}(\frac{\partial F}{\partial u_{xx}}) + \frac{\mathrm{d^2}}{\mathrm{d}y^2}(\frac{\partial F}{\partial u_{yy}})$$
However, I'm not sure about what to do with the partial derivative with respect to both $x$ and $y$. Thus far, I've found that: $$\frac{\partial F}{\partial u} = 2u, \frac{\mathrm{d}}{\mathrm{d}x}(\frac{\partial F}{\partial u_x}) = 2u_{xx}, \frac{\mathrm{d}}{\mathrm{d}y}(\frac{\partial F}{\partial u_y}) = 2u_{yy}, \frac{\mathrm{d^2}}{\mathrm{d}x^2}(\frac{\partial F}{\partial u_{xx}}) = 2u_{xxxx}, \frac{\mathrm{d^2}}{\mathrm{d}y^2}(\frac{\partial F}{\partial u_{yy}}) = 2u_{yyyy}$$If anyone could help and could point out any mistakes I've made thus far, that would be much appreciated.