This is Lemma 5.2 in Lee's Riemannian manifolds book. Given a Riemannian manifold $(M,g)$ with a connection that is compatible with $g$, I cannot show why the following holds for vector fields $Y,Z$ along any curve $\gamma$: $$\frac{d}{dt}g(Y,Z) = g(D_tY,Z) + g(Y,D_tZ)$$
Since $Y$ and $Z$ are vector fields along a curve, does this mean that when you evaluate them at some $t \in I$ that you actually use $Y(\gamma(t))$?
Because I thought you could just use that $$\frac{d}{dt}g(Y,Z)(\gamma(t)) = \frac{d}{dt}(g(Y,Z) \circ \gamma)(t) = \dot{\gamma}(t)\left(g(Y,Z)\right) = \nabla_{\dot{\gamma}(t)}g(Y,Z) = D_t g(Y,Z)$$
And the result would follow from $\nabla$ being compatible with $g$.
Can someone please validate this? I'm not sure whether I have understood the definition of vector fields along curves correctly.