Lie Derivative of the Jacobian?

$X,Y$ smooth manifolds.
Given a map $f:X\longrightarrow Y$. Let’s say for simplicity that $f$ is a diffeomorphism and defined everywhere.
Let $v,w$ be smooth vectorfields on $X$ and $g\in\mathcal C^\infty(Y)$ a testfunction on $Y$.
The Jacobian $Df|_x$ of $f$ is a linear map $Df|_x:T_xX\longrightarrow T_{f(x)}Y$ defined by
\begin{equation}
Df|_x w|_x [g]=w[g\circ f](x)
\end{equation}

I’m wondering whether one can define a Lie derivative of the Jacobian along some vectorfield $v$?

It should look like this:
\begin{equation}
\mathcal L_v Df:=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\Big[ \big(\Phi_{-\epsilon}\big)_*Df\big|_{\Phi_\epsilon(x)}-Df\big|_{x} \Big]
\end{equation}
where $\Phi_t(x)$ is the solution curve along vector field $v$ starting at $x$. It is a diffeomorphism.

$\big(\Phi_t\big)_*$ should be understood as some sort of pushforward for Jacobians. A natural way to do this is would be:
\begin{equation}
\Phi_* Df=D(f\circ \Phi^{-1})
\end{equation}

In local coordinates we would get
\begin{equation}
\big(\mathcal L_v Df\big)^k_i=\sum_j\frac{\partial}{\partial x^i}\Big(v^j\frac{\partial f^k}{\partial x^j}\Big)\Big|_{x}
\end{equation}
This would be very convenient because it preserves the linearity of the Lie derivative:
\begin{equation}
\mathcal L_v \big(Df w\big)=\big(\mathcal L_v Df\big)w + Df\big(\mathcal L_v w)
\end{equation}

But I’m not sure whether there is something like a pushforward of the Jacobian or indeed a Lie derivative of the Jacobian. Maybe someone can point me to a source?

Thanks a lot!

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