I have been reading about Jacobi fields in do Carmo's book "Riemannian geometry", and I have a little question.
Let $(M,g)$ be a riemannian manifold and let $p\in M$ and $v\in T_p M$ such that $\exp_p v$ is defined.
It begins with a parametrized surface
$$f(t,s)=\exp_{p}tv(s)\qquad 0\leq t\leq 1,\, -\varepsilon\leq s\leq\varepsilon$$
where $v(s)$ is a curve in $T_p M$ such that $v(0)=v$ and $v'(0)=w\in T_v(T_p M)$.
By the Gauss lemma we have
$$(d\exp_p)_v w=\frac{\partial f}{\partial s}(1,0)$$
So far so good.
Then it goes on by saying
"It is convenient to extend our objective slightly and study the field $$(d\exp_p)_{tv}(tw)=\frac{\partial f}{\partial s}(t,0)$$ along the geodesic $\gamma(t)=\exp_p(tv)$."
Since $\gamma$ is a geodesic, we have $\nabla_{\dot\gamma}\dot\gamma=0$. But in the book they claim that we also have
$$\nabla_{\dot\gamma(t)}\frac{\partial f}{\partial t}(t,s)=0$$
for all $(t,s)$. Why is that?
I see why this holds for $s=0$, since $v(0)=v$.
But why would that be for $s\neq 0$?
Any help would be very much appreciated!