$\rho$ is a probability measures on $\Omega$ and $\epsilon$ is a given energy as below:
$$\epsilon (\rho) = \int_{\Omega} \Big[\alpha U(\rho(x)) + V(x) \rho(x)\Big] \, \mathrm dx + \frac{1}{2}\int_{\Omega \times \Omega} \Big[W(x-y) \rho(x) \rho(y)\Big] \, \mathrm dx \,\mathrm dy.$$
My question is how to prove the derivate of $\epsilon$ with respect to $\rho$ is
$$\frac{\delta \epsilon}{\delta \rho} = \alpha U'(\rho) + V + W\rho.$$
This question is from the paper I'm reading: Primal Dual Method for Wasserstein Gradient Flows. https://link.springer.com/article/10.1007/s10208-021-09503-1.