Differential of the flow of a vector field

Question

Let $M$ be a compact manifold, and let $X$ be a vector field. Then its flow, $\varphi$, generates a $1$-parameter group of diffeomorphisms $\{\varphi_t\}$. For each $t$, we have a map $\varphi_t:M\to M$. Thus, for $p\in M$, the differential $\mathrm d(\varphi_t)_p:T_pM\to T_{\varphi_t(p)}M$ is defined. Is there a general formula for $\mathrm d (\varphi_t)_p(v),v\in T_pM$?

I am particularly interested in the case when $X=\mathrm{grad}\,f$, where $f$ is a smooth function and the gradient is determined using a Riemannian metric $g$ on $M$ and when $p$ is a critical point of $f$. Then $X$ vanishes at $p$ and the flow has $p$ as a fixed point, so $\mathrm d(\varphi_t)_p:T_pM\to T_pM$ is an endomorphism.

Answer 1

I'm not sure if this answers your question, but it may be a step in the right direction. There is a nice equation called the first variation equation for the flow giving you a differential equation for the object you're interested in if you can localize in $\Bbb R^n$: $$\frac d{dt} d(\varphi_t)(p) = dX(\varphi_t(p))d(\varphi_t)(p), \quad d(\varphi_0)(p) = \text{id}.$$ See also this SE question. So perhaps you know enough to get information by integrating this differential equation from $0$ to $t$.