Does a Chain Rule Exist for Functional Derivatives?

[shoehorn]

Does there exist a chain rule for functional derivatives? For example, in ordinary univariate calculus, if we have some function $y=y(x)$ then the chain rule tells us (loosely) that

$$\frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}.$$

Now suppose that we have a functional $F[f;x)$ of some function $f(x)$. The functional derivative of $F[f;x)$ is denoted

$$\frac{\delta F[f;x)}{\delta f(y)}.$$

However, suppose that $f$ is itself a functional of a function $g(x)$. Can I then write

$$\frac{\delta}{\delta f} = <br /> \frac{\delta g}{\delta f} \frac{\delta}{\delta g}?$$

 

[DeadWolfe]

http://en.wikipedia.org/wiki/Chain_rule#Proof_of_the_chain_rule

The proof looks like it should work for any Frechet derivative.