Let F(g(f)) be a functional that sends functions of a vector variable ( from n -dimensional vector space) to R.
g is some function of scalar valued functions f .
It can be shown using vector calculus that the first (order) functional derivative $\partial F$ is equal to the partial derivative of g with respect to f minus the divergence of the gradient of g .
https://en.m.wikipedia.org/wiki/Functional_derivative
$F=\int f ^{4} dx^{4}$
$\partial F= \frac{\partial{g}}{\partial{f}} - \nabla \cdot\nabla g(f)$
Divergence of the gradient should be equal to 4 , and if I plug in the above functional into the formula on wiki it should work just as well for 4d and I get the result :$ \partial F=−12f^{3}$
Can I just plug this into the formula again and get the second derivative of the functional?