  In Combinatorics Stirling numbers may be defined in several ways. One such
definition is given in [1], where an extensive consideration of Stirling
numbers is presented. In this paper an alternative definition of Stirling
numbers of both kind is given. Namely, Stirling numbers of the first kind
appear in the closed formula for the n-th derivative of ln x. In the same way
Stirling numbers of the second kind appear in the formula for the n-th
derivative of f(e^x), where f(x) is an arbitrary smooth real function. This
facts allow us to define Stirling numbers within the frame of differential
calculus. These definitions may be interesting because arbitrary functions
appear in them. Choosing suitable function we may obtain different properties
of Stirling numbers by the use of derivatives only. Using simple properties of
derivatives we obtain here three important properties of Stirling numbers.
First are so called two terms recurrence relations, from which one can easily
derive the combinatorial meaning of Stirling numbers. Next we obtain expansion
formulas of powers into falling factorials, and vise versa. These expansions
usually serve as the definitions of Stirling numbers, as in [1]. Finally, we
obtain the exponential generating functions for Stirling and Bell numbers. As a
by product the closed formulas for the $n$-th derivative of the functions
f(e^x) and f(ln x) are obtained.
