I want to find
$\arg\!\min_u \frac 1 2 \|u-f\|^2 + \frac 1 2 \|d_x - u_x\|^2 +\frac 1 2 \| d_y - u_y \|^2 := \arg\!\min_u F(u,u_x,u_y) $
where $u$ is a function of $x$, $y$ and $u_x$, $u_y$ are partial derivatives.
I saw that it uses variational derivative, but the wikipedia deals only with the integral form (namely, to minimize $\int L$, we have $L_u - \frac{\partial} {\partial x} L_{u_x} - \frac{\partial} {\partial y} L_{u_y}= 0$).
How about the case of the above problem? I tried to follow the proof of integral case, but there was some problem when the proof uses integral by parts.
I guess that the answer will be $F_u - \frac{\partial} {\partial x} F_{u_x} - \frac{\partial} {\partial y} F_{u_y}= 0$. Is it true? How can I prove it and where can I find related materials?