this general theory is its application to variational and perturbative analysis, and we will do this in upcoming papers.

Like many very general theorems, our main abstract result, Theorem \[thm:characterization\], is quite simple. It states that *the set of possible derivatives $\gamma'(0)$ of differentiable curves $\gamma\colon [0,+\infty)\to C$ emanating from a point $p=\gamma(0)$ in a closed, convex subset $C$ of a locally-convex topological vector field, coincides with the sequential closure of the tangent cone of $C$ at $p$;* see Section \[sec:characterization\] for definitions and for the precise statement. We also give reformulations in Corollaries \[cor:normed\] and \[cor:useful\] that work around the issue of taking the sequential closure, and henced may be more amenable to practical use.

In the second part of the paper, we consider families of measures varying in a weakly differentiable way: we merely ask for the map $t\mapsto \int \phi\,d\mu_t$ to be differentiable at 0 for all $\phi\in C^\infty_c$. Such families arise naturally in many parts of mathematics, including the theory of the heat equation, stochastic dynamics and <span style="font-variant:small-caps;">spde</span>s, and the continuity equation in mass transportation, to name a few.

As we show in Examples \[ex:deltas\], \[ex:uniform\] and \[ex:explicit\], the derivative of a family of positive 