The Wasserstein space can be given a family of geometric tangent cones built from geodesics. These tangent cones form a subset of measures on the tangent bundle, and most often, a strict subset. This talk addresses the following problem: can we know if an element belongs to the tangent cone by looking at the directional derivative of the Wasserstein distance along it? We give some context around this question, the currently available results and some open developments.