Let $M$ be a simply connected complete Kähler manifold and $N$ a closed complete totally geodesic complex submanifold of $M$ such that every minimal geodesic in $N$ is minimal in $M$. Let $U_\nu$ be the unit normal bundle of $N$ in $M$. We prove that if a distance function $\rho$ is differentiable at $v\in U_\nu$, then $\rho$ is also differentiable at $-v$.