A didactical exposition of the classical problem of the trajectory determination of a body, subject to the gravity in a resistant medium, is proposed. Our revisitation is aimed at showing a derivation of the problem solution which should be as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates. A central role in our analysis is played by the so-called "chain rule" for derivatives, which is systematically used to remove the temporal variable from Newton's law in order to derive the differential equation of the Cartesian representation of the trajectory, with a considerable reduction of the overall mathematical complexity. In particular, for a resistant medium exerting a force quadratic with respect to the velocity our approach leads to the differential equation of the trajectory, which allows its Taylor series expansion to be derived in an easy way. A numerical comparison of the polynomial approximants obtained by truncating such series with the solution recently proposed through a homotopy analysis is also presented.