Functional derivative of four-gradient

Question

I am begining a course in QFT, and am starting with the topic of functional derviatives. In the problem set given, we are asked to calculate the functional derivative of $$F[\phi] = \partial_{x}\phi(x).$$ I am kind of at a loss as to what this means.

From what I understand, $$\partial_{x} = (\partial_{t},-\partial_{x},-\partial_{y},-\partial_{z}),$$

so i suppose the $x$ in $\phi(x)$ actually implies a dependence on the four-position vector. So, am i supposed to calculate a functional derivative for each component?

Any help is appreciated.

Answer 1

I think you are meant to just consider functions of one variable, so your multidimensional gradient is barking up the wrong tree. $\partial_x = {d\over dx}$ in this context.

If this is part of a QFT course, you are meant to use $$ \frac{\delta \phi(x)}{\delta \phi(y)} = \delta(x-y), ~~\leadsto \\ \frac{ \delta (\partial_x\phi(x))}{\delta \phi(y)} = \partial_x \delta(x-y) ~. $$