Let $(M,g)$ be a Riemannain manifold and let $p\in M$. Let $\gamma:[0,1] \to M$ be a smooth curve and let $p \notin \gamma([0,1])$. Assume further that for each $t \in [0,1]$ there is a unique (unit speed) geodesic from $p$ to $\gamma(t)$ with initial speed $v(t) \in T_p M$. Is $t \mapsto v(t)$ differentiable? If so, what is the derivative of $v$ (in terms of $\gamma, g, p$)?