Path derived numbers and path derivatives of continuous functions with respect to continuous systems of paths

V. Jarnik showed that a typical continuous function on the unit interval [0, 1] has every extended real number as a derived number at every point of [0, 1]. In this paper we classify the derived numbers of a continuous function and study the likelihood of Jarnik's Theorem for path derived numbers of a continuous system of paths. We also provide some results indicating that some of the nice behaviors of first return derivatives are shared by path derivatives of continuous functions when the path system is continuous.