The functional derivative we talk about here is not the same as the variation used widely in Lagrangian formalism — although they look so much alike formally. To make sure you will not mistake them, we first illustrate what is the variation of a functional.
The variation of a functional \(F[\phi]\) whose domain is a Banach space \(B\) (a linear space where you can talk about the “length” — norm of its element) is usually implemented as Gâteaux derivative
which very much resembles the directional derivative in ordinary Calculus. Please notice that the \(\eta\) here is just an arbitrary element of the Banach space, and in its application it is any physical field.
The variation is the “derivative” of the argument \(\phi\) itself, without any designation of it value at some point \(\phi(x)\); while functional derivative is defined differently. The functional derivative of \(F[\phi(x)]\) with respect to value \(\phi(y)\) is
Therefore, you shall now see the differences between the variation and the functional derivative — the variation performs “derivative” on \(\phi\) itself, while the functional derivative does so on the value \(\phi(y)\). And thus a delta function has to be invoked to deal with the different points \(x\) and \(y\). You will see in chapter Quantum Theory that it is this delta function which brings the infinities into the S-matrix.