A Universal Differentiability Set, UDS for short, is a set such that every Lipschitz function is differentiable in at least one point of the set. The existence of measure zero UDS, which is strictly related to Rademacher's theorem, has already been established in the Euclidean setting and for certain classes of Carnot groups. The key technique underlying the construction of UDS is a result that relates maximality of a directional derivative for a Lipschitz function with differentiability. In this seminar we will investigate this connection and we will try to establish how things change when we work in a nonlinear setting. To do so we will focus our attention on the Laakso space, one of the best known and most interesting instances of a metric measure space which admits a differentiable structure but not a linear structure. An introduction to Laakso spaces and their structure will be provided during the seminar.