Directed polymers are a model of paths that are arranged with a tendency to minimize an energy with contributions from a random potential and self-interaction. In these models the shape function at slope v corresponds to the average action accrued from paths traveling with an average slope v, as the length of the path tends to infinity. While it is not difficult in general contexts to show that such the shape function exists and is convex, many of its properties, like whether it is strictly convex or differentiable everywhere, are not well understood. In this talk I will discuss a model that is discrete in time and continuous in space in which we are able to prove differentiability of the associated shape function. Our techniques work in the zero and positive temperature regimes and for a broad class of environment potentials and kinetic energies. Based on joint work with Yuri Bakhtin.