I have a Lagrangian of the form $\mathcal{L}(x,f)=[s(x)-\int_a^b A(x,x')f(x')dx']g(x)$, where $a,b$ are constants, and $g(x),s(x)$ and the kernel $A(x,x')$ are given .
I am interested in computing the variational derivative $\frac{\partial \mathcal{L}}{\partial f}$ for use in the Euler-Lagrange equation. How do I proceed?
Hint: Vary the action functional $$ S[f]~:=~\int_a^b\! dx ~{\cal L}.$$ The stationary condition becomes $$ \forall x^{\prime} \in [a,b]:~~ \int_a^b\! dx~g(x) A(x,x^{\prime})~=~0 . $$