Let $(M,g)$ be a Riemannian manifold and consider the metric
$$d(p,q) = \inf \{ l(\gamma) | \gamma: [0,1] \to M \text{ piecewise smooth with } \gamma(0) = p, \gamma (1) = q \}$$ where $l(\gamma)$ is the length of a curve.
In Eucliden space $\mathbb{R}^n$, this metric coincides with the standard euclidean metric and in particular we have that geodesics are straight lines, that is a geodesic takes the form $$ c(t) = p + vt.$$ That is, in particular if we consider a unit speed geodesic it has
$$ \frac{d}{dt} c(t) = \frac{q-p}{||q-p||} $$ for some $q$, but then also (for $t \in [0,1)$ ) $$ \frac{d}{dt} c(t) = \frac{q-c(t)}{||q-c(t)||} = -\operatorname{grad} d(q,\cdot)_{c(t)}. $$
In my intuition, the same should also hold locally for geodesics on an arbitrary Riemannian manifold, that is in a normal neighborhood around some point $p$ , do we have that a geodesic $c(t)$ fulfills
$$ \frac{d}{dt} c(t) = -\operatorname{grad} d(q,\cdot)_{c(t)}?$$