One-sided derivative of distance to a compact set

We give a complete and self-contained proof of folklore theorem which says that in an Alexandrov space the distance between point γ(t) on geodesic γ compact set K is right-differentiable function t. Moreover, value this right-derivative given by negative cosine minimal angle any shortest path to (Theorem 4.3). Our treatment serves as general introduction metric geometry relies only basic elements, such comparison triangles upper angles.