  Higher order derivatives of functions are structured high dimensional objects
which lend themselves to many alternative representations, with the most
popular being multi-index, matrix and tensor representations. The choice
between them depends on the desired analysis since each presents its own
advantages and disadvantages. In this paper, we highlight a vectorized
representation, in which higher order derivatives are expressed as vectors.
This allows us to construct an elegant and rigorous algebra of vector-valued
functions of vector variables, which would be unwieldy, if not impossible, to
do so using the other representations. The fundamental results that we
establish for this algebra of differentials are the identification theorems,
with concise existence and uniqueness properties, between differentials and
derivatives of an arbitrary order. From these fundamental identifications, we
develop further analytic tools, including a Leibniz rule for the product of
functions, and a chain rule (Fa\`a di Bruno's formula) for the composition of
functions. To complete our exposition, we illustrate how existing tools (such
as Taylor's theorem) can be incorporated into and generalized within our
framework of higher order differential calculus.
