Consider the functional $F$ defined via the integral $$ F(\mu)=\int_0^\ell\int_0^\ell f(s,t)\mu(s)\mu(t)\,ds\,dt. $$ How would I differentiate this with respect to $\mu$?
I realize that this has something to do with the functional derivative, but I can't seem to put it all together here. There appear to be other questions that deal with a similar topic, but this one seems to be complicated by the fact that each factor of $\mu$ is dependent on a different variable.
For simplicity, just assume that $f$ and $\mu$ are as nice as we would need them to be.
Replace $\mu$ with $\mu+\eta$, thinking of $\eta$ as small perturbation. Since $F$ is just a quadratic function in $\mu$, only algebra is needed to isolate the linear term with respect to $\eta$. This term is the derivative. $$\nabla F(\mu)(\eta)=\int_0^\ell\int_0^\ell f(s,t)\nu(s)\mu(t)\,ds\,dt + \int_0^\ell\int_0^\ell f(s,t)\mu(s)\nu(t)\,ds\,dt\tag{1}$$ meaning that the derivative of $F$ at $\mu$ is the linear functional sending each $\eta$ to the right hand side of $(1)$.