It has also the name of variational derivative, and it is the equivalent to the derivative of a function.

Intuition and working definition

We proceed with analogy to the ordinary derivative at x

$$ f(x) \to \left.\frac{d}{dy} f(y)\right|_{y=x} \equiv \lim_{\varepsilon\mapsto 0 } \frac{f(x+\varepsilon)-f(x)}{\varepsilon} = \frac{\partial}{\partial \varepsilon}f(x+\varepsilon) $$ where the last expression is equivalent, and will help us in the next definition, $\varepsilon$ represents the variation around the point $x$.

For a functional, we will denote the derivative with $\delta$, as representation of variation. As we know how to derive functions of "numbers", we will transform our funtional of $q$ to a "normal" function for $\varepsilon$ with the map $F[q]\mapsto F[q+\varepsilon h]$ where $h$ is called a test function (analogous to a vector on multivariate calculus), then derive:

$$ S[q] \to \delta F[q|\phi] = \frac{\partial}{\partial \varepsilon}F[q+\varepsilon\phi] $$

Formal Definitions

Frechet derivative on Banach spaces, and more generally Gatheaux derivative on locally convex spaces …