# Variational Derivative Relationships

I am interested in a free-energy functional given by: $$E[\phi] = \int f(\phi) - \frac{\epsilon^2}{2}|\nabla\phi|^2 \ d\vec{x} = \int F(\phi)$$ My understanding of variational derivatives is quite weak, and so, I am wondering what to make of: $$\frac{\delta E}{\delta (1-\phi)}=\ ????$$ From what I understand, the variational derivative with respect to $$\phi$$ can be calculated as: $$\frac{\delta E}{\delta \phi} = \frac{\partial F}{\partial \phi}-\nabla\cdot\frac{\partial F}{\partial \nabla\phi} = f'(\phi)-\epsilon^2\nabla^2\phi$$ My question then: is it true that: $$$$\pmb{\frac{\delta E}{\delta (1-\phi)}=-\frac{\delta E}{\delta \phi}}\ ????$$$$ If it matters, in this context $$\phi=\phi(\vec{x},t)$$ represents the composition of component A in a binary mixture. (Hence, $$1-\phi$$ is the composition of component B).

As a simple explanation, changing $$1-\phi$$ by $$\epsilon \psi$$ amounts to changing $$\phi$$ by $$-\epsilon \psi$$, so $$E$$ responds as if you had done that change to $$\phi$$.