Differential of the exponential map

Question

Let $M$ be a compact Riemannian manifold without boundary. It is well known that for every $p \in M$ there exists a neighborhood $V_p$ of $0_p \in T_pM$ s.t. the map $$ \exp_p : V_p \to \exp_p(V_p) $$ is a diffeomorphism. Is its differential a continuous mapping? I would like to say that the quantities $$ \sup_{u \in U } \| \text{d}\exp(u) \| \quad \sup_{u \in U } \| \text{d}\exp^{-1}(u) \| $$ are finite, where $$ U = \{ u \in TTM \mid \|u\| = 1 \} $$. This would immediately follow if $\text{d} \exp$ and its inverse are continuous thanks to the compactness of $U$ (I am endowing $TM$ with the Riemannian metric induced by $M$).