  We prove a formula for the $n$-th derivative of the period function $T$ in a
period annulus of a planar differential system. For $n = 1$, we obtain Freire,
Gasull and Guillamon formula for the period's first derivative \cite{FGG}. We
apply such a result to hamiltonian systems with separable variables and other
systems. We give some sufficient conditions for the period function of
conservative second order O.D.E.'s to be convex.
