Let $(M, g)$ be a smooth Riemannian manifold, $p \in M$, and $\exp_P$ the exponential map at the point $P$: $\exp_P: T(P) \to M$

It seems clear to me that $\exp_P$ is smooth on $U \setminus \{0\}$, where $U$ is a neighborhood of the origin $0 \in T(P)$, because $\exp_P$ is defined from the geodesics, which are solutions of a system of ordinary differential equations.

But is it true that $\exp_P$ is also always smooth at the origin $0 \in T(P)$?

I am not interested for the moment on the ways to calculate these high order derivatives, but just on the theoretical question about their existence.

Thank you!