In multivariable calculus we learn that partial derivatives are commutative: i.e. $\partial _{xy}F=\partial_{yx}F$
When dealing with multiple functional derivatives this doesn't seem to be the case.
Consider the simple functional equation: $\mathcal{I}[f]=\int f(x)dx$
It would seem that if we compute the functional derivative: $\large\frac{\delta^2\mathcal{I}[f^3]}{\delta f(x_0)\delta f(x_1)}$ the same rules of commutivity that apply in multivariable calculus are not present. Depending on the order we take the two derivatives, we would either get $6[f(x_0)]$ or $ 6[f(x_1)]$.
My question is obvious: which is the proper answer? It would seem to me that the derivative with respect to $f(x_1)$ would be computed last and thus the ladder of the two would be correct.