On differential operators in white noise analysis

White noise analysis is formulated on a general probability space which is such that (1) it admits a standard Brownian motion, and (2) its sigma -algebra is generated by this Brownian motion (up to completion). As a special case, the white noise probability space with time parameter being the half-line is worked out in detail. It is shown that the usual differential operators can be defined on the smooth, finitely based functions of at most exponential growth via the chain rule, without supposing the existence of a linear structure (or translations) on the underlying probability space.