  Several versions of It\^{o}'s formula have been obtained in the context of
the functional stochastic calculus. Here, we revisit this topic in two ways.
First, by defining a notion of derivative along a functional, we extend the
setting of the (semimartingale) functional It\^{o}'s formula and corresponding
calculus. Second, for L\'{e}vy processes, an optimal local-time based It\^{o}'s
formula is obtained. Some quick applications are then given.
