In this talk I will discuss sharp differential inequalities for the isoperimetric profile function in spaces with Ricci bounded from below, and with volumes of unit balls uniformly bounded from below. After that, I will highlight some of the consequences of such inequalities for the isoperimetric problem. After a short introduction about the notion of perimeter in the metric measure setting, I will pass to the motivation and statement of the sharp differential inequalities on Riemannian manifolds. Hence, I will discuss the proof, which builds on a non smooth generalized existence theorem for the isoperimetric problem (after Ritoré-Rosales, and Nardulli), and on a non smooth sharp Laplacian comparison theorem for the distance function from isoperimetric boundaries (after Mondino-Semola). At the end I will discuss how to use such differential inequalities to study the behaviour of the isoperimetric profile for small volumes. This talk is based on some results that recently appeared in a work in collaboration with E. Pasqualetto, M. Pozzetta, and D. Semola. Some of the tools and ideas exploited for the proofs come from other works in collaboration with E. Bruè, M. Fogagnolo, and S. Nardulli.