I am trying to make sense of functional derivatives and have a couple of questions bothering me:
Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.
By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$\frac{\delta F}{\delta G} = \int dt \frac{\delta F[X]}{\delta X(t)} \frac{\delta X(t)}{\delta G[X]} = \int dt \frac{\delta F[X]}{\delta X(t)} \left(\frac{\delta G[X]}{\delta X(t)} \right)^{-1} $$ Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?
How would one calculate the functional derivatives $\frac{\delta F[X]}{\delta X(t)}$ and $\frac{\delta G[X]}{\delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?