Radon measures is rarely a signed Radon measure, but instead in general it is a distribution, that is, a continuous real-valued functional on the space of $C^\infty$ functions. We apply our general theory to characterize in Theorem \[thm:families\] the distributions that arise as derivatives of families of probabilities on a smooth manifold; we do this in Section \[sec:families\]. Our result states that *given a measure $\mu$, the distributions $\eta$ that arise as derivatives of differentiable families $(\mu_t)_{t{\geqslant}0}$ starting at $\mu_0=\mu$ are precisely those that satisfy that $\langle\eta,f\rangle{\geqslant}0$ for all nonnegative functions $f\in C^\infty_c$ that vanish on the support of $\mu$.*

In Section \[sec:examples\] we give some important remarks and examples that should aid in understanding the content of the statement of the characterization. We also give a characterization of the existence of curves with two-sided derivatives; this is Corollary \[cor:two-sided\]. Then in Section \[sec:masstransport\] we consider the continuity equation from mass transportation, namely $$\frac{d\mu_t}{dt}+\operatorname{div}(v_t\mu_t)=0,$$ in a very weak sense, and we use it to find elements of a Colombeau algebra that provide a meaning to the concept of “direction of the flow,” that is, we find an object that has a role similar to the classical vector field $v_t$, but 