  This contribution proposes a new formulation to efficiently compute
directional derivatives of order one to fourth. The formulation is based on
automatic differentiation implemented with dual numbers. Directional
derivatives are particular cases of symmetric multilinear forms; therefore,
using their symmetric properties and their coordinate representation, we
implement functions to calculate mixed partial derivatives. Moreover, with
directional derivatives, we deduce concise formulas for the velocity,
acceleration, jerk, and jounce/snap vectors. The utility of our formulation is
proved with three examples. The first example presents a comparison against the
forward mode of finite differences to compute the fourth-order directional
derivative of a scalar function. To this end, we have coded the finite
differences method to calculate partial derivatives until the fourth order, to
any order of approximation. The second example presents efficient computations
of the velocity, acceleration, jerk, and jounce/snap. Finally, the third
example is related to the computation of some partial derivatives. The
implemented code of the proposed formulation and the finite differences method
is proportioned as additional material to this article.
