Chain rule with functional derivative

[jostpuur]

This is supposedly the chain rule with functional derivative:

$$\frac{\delta F}{\delta\psi(x)} = \int dy\; \frac{\delta F}{\delta\phi(y)}\frac{\delta\phi(y)}{\delta\psi(x)}$$

I have difficulty understanding what everything in this identity means. The functional derivative is usually a derivative of a functional with respect to some function, like in the term

$$\frac{\delta F}{\delta\phi(y)} := \lim_{\epsilon\to 0} \frac{1}{\epsilon}\big( F(\phi + \epsilon \delta_y) - F(\phi)\big),$$

but isn't the term

$$\frac{\delta\phi(y)}{\delta\psi(x)} := ?$$

now a derivative of a function with respect to another function?

 

[jostpuur]

It could be I understood this. If $\phi$ depends on $\psi$ somehow, that means that every $\psi$ can be mapped into a set of functions $\{\phi\}$, $\psi\mapsto\phi_{\psi}$, then with a fixed y there's the natural functional G, $G(\psi) = \phi_{\psi}(y)$.