Is the distance function to $p$ differentiable or smooth at $q$ if there is a unique minimizing geodesic connecting them?

Question

Let $(M,g)$ be a complete smooth Riemannian manifold. Assume that for two points $p$ and $q$ in $M$, there is a unique minimizing geodesic connecting them. Denote the distance function to $p$ by $d_p(x)$. Then is $d_p(x)$ differentiable at $q$? If this is true, then how to prove it? Furthermore, is $d_p(x)$ smooth at $q$?