If $F$ is a function of $x$ and $y$, and $F_x$ denotes $\dfrac{\partial F}{\partial x}$, how would you compute $\dfrac{\partial F_x}{\partial F}$? Can I use the chain rule to write this as: $\dfrac{\partial F_x}{\partial x}\dfrac{\partial x}{\partial F}$?
Edit:
I encountered this problem while trying to write the Euler-Lagrange (E-L) equation for the functional $J(I) = \int\int_\Omega \sqrt{I_x(x, y)^2 + I_y(x, y)^2} dxdy$, which is of the form $J(z) = \int\int_\Omega F(x,y,z,z_x,z_y) dxdy$. The E-L equation for such a functional is $\dfrac{\partial F}{\partial z} - \dfrac{\partial}{\partial x}\dfrac{\partial F}{\partial z_x} - \dfrac{\partial}{\partial y}\dfrac{\partial F}{\partial z_y} = 0$. I do not know how to compute $\dfrac{\partial F}{\partial I}$ because it involves partial derivative terms like the one mentioned in the question title.
For every differentiable $H$ and every $I$ such that $I_x^2+I_y^2\gt0$ everywhere on $\Omega$, $$\langle\partial J(I),H\rangle=\iint_\Omega\frac{H_x\cdot I_x+H_y\cdot I_y}{\sqrt{I_x^2+I_y^2}}$$