Let A be the set of all isomorphism classes of atomic species, let K be a binomial half-ring and K^- its rational closure. The differential half-ring K[[A]] of all K-species in the sense of Yeh is a combinatorial and algebraic extension of the half-ring K[[X]] of all formal power series in one indeterminate X. Using the operation of substitution in K[[A]] and the Q-species X^{^} of "pseudo-singletons" we study two new notions: the combinatorial directional derivative of a K-species in the direction of another K-species and Taylor expansions in K[[A]]. The use of K^--species is essential here. We show, along the way, certain similarities and differences between these new notions and their classical analogues in K[[X]]. Tables are given for small cardinalities.

The paper has been finally published under the same title in Discrete Math. 79 (1990), 279-297.