Consider a Riemannian manifold $M$, a unit speed geodesic $\gamma\colon[0,l]\to M$ and a parallel unit vector field $E$ along $\gamma$. Let $f\colon [0,l]\to \mathbb{R}$ be some smooth function. Consider $t\mapsto f(t)E(t)$ as a path in $TM$.
Then the velocity vector $\dot{f(t)E(t)}$ should be $\dot{\gamma}(t)+f^{\prime}(t)E(t)$, but I don't understand the second term in this decomposition.
Does anyone know a formal proof of this?