Derivative of the flow for ODEs on manifolds

As was pointed out by Deane Yang and Igor Khavkine in the comments, this feels like a fact that should be "looser" than Riemannian geometry. Indeed, as I will show below, your formula makes sense in the broader setting of smooth manifolds equipped with a volume form. I don't know a reference for this fact.

If $(M, \Omega)$ is a manifold equipped with volume form, ${\bf V}\in \operatorname{Vect}(M)$ is a vector field on $M$, and ${\bf X}_t:M\to M$ is the flow of ${\bf V}$, then:

1) One has $J_t\in C^\infty(M)$, (i.e., a smooth $J:\mathbb{R}\times M\to M$), with $J_0\equiv 1$, given by, $$J_t(x)=(\Omega_x)^{-1}\otimes \Omega_{{\bf X}_t(x)}\otimes \Lambda^n(d{\bf X}_t|_x),$$ well-defined since $\Lambda^n(d{\bf X}_t|_x):\Lambda^n( T_xM)\to \Lambda^n (T_{{\bf X}_t(x)}M)$ as Igor said.

2) One has a well-defined $\operatorname{div}({\bf V})$, namely (up to a sign, I forget whether $\pm$) $$\operatorname{div}({\bf V})=d(\iota_{\bf V}\Omega)/\Omega.$$

So the "hopefully-an-identity" $$\left.\frac{d}{dt}\right\rvert_{t=0}J_t=\operatorname{div}({\bf V})$$ (for simplicity I state just the $t=0$ formula) is well-formed. You can then prove it by choosing local co-ordinates in which $\Omega\equiv 1$. (There is surely an intrinsic proof, too, but I can't think of one now.)

Afterthought, 30 Oct: I imagine the intrinsic proof consists of identifying both sides with the Lie derivative $(\mathscr{L}_{\bf V}\Omega)/\Omega$.

Update, 30 Oct, in response to questions of OP in comments: If I calculate correctly, then yes, the formulas \begin{align*} \frac{dJ_t}{dt}(x)&=\operatorname{div}({\bf V})|_{{\bf X}_t(x)} J_t(x)\\ J_t(x)&=\exp\left(\int_0^t\operatorname{div}({\bf V})|_{{\bf X}_s(x)}ds\right) \end{align*} are also correct.

And yes, although the formula for "$\tfrac{d}{dt}\left[\det\left(d{\bf X}_t\right)\right]$" depends only on the volume form, I think one needs a Riemannian metric $g$ in order to make sense of "$\tfrac{d}{dt}\left[d{\bf X}_t\right]$" (really, the $g$-covariant derivative $D_t$ of the section $d{\bf X}_t$ of the bundle $T_x^*M\otimes TM$ along the curve ${\bf X}_t(x)$). Probably $$D_t\left[d{\bf X}_t\right]|_{t=0}=\nabla {\bf V},$$ where $\nabla$ is the Levi-Civita connection of $g$. (Please check this calculation before relying on it!)

I think you may be interested in this remarkable, recent paper by E. Brué and D. Semola (see in particular Theorem 3.11 which answers to your question in a much more general setting).