= Functional derivative
{title2=$\delta F/\delta\phi$}
{wiki}
A <functional derivative> represents the linear first-order response of a functional to a perturbation of its input <function>. With a specified integration measure and admissible perturbations,
$$
F[\phi+\epsilon h]-F[\phi]=\epsilon\int\frac{\delta F}{\delta\phi(x)}h(x)\,dx+o(\epsilon).
$$
For $F[\phi]=\int\phi(x)^2dx$, direct expansion gives $\delta F/\delta\phi=2\phi$. For $F[\phi]=\frac12\int|\phi'|^2dx$, <integration by parts> gives $-\phi''$ when the <boundary term> vanishes under the chosen perturbation conditions. The <boundary conditions> and <function space> are therefore part of identifying the <derivative>, rather than optional details.
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