Statement of the Problem
Let $x=x(t) \in \mathbb{R^n}$ denote Lagrangian spatial coordinates, where $t\in \mathbb{R}_{>0}$ is time. Let $x_0 \in \mathbb{R^n}$ denote the normal Euclidean coordinates.
We are concerned with the derivation of a formula involving these coordinates and the velocity field $v(x,t) = (v_1, v_2, ..., v_n) : \mathbb{R^n} \times \mathbb{R}_{>0} \to \mathbb{R^n}$. In particular, we claim that
$$ \frac{\text{d}}{\text{d}t} \frac{\partial x}{\partial x_0} = \frac{\partial x}{\partial x_0} \nabla \cdot v, $$
for all $n \geq 2$.
My Work So Far
It is easy to show that this formula holds explicitly in the $2$- and $3$-dimensional cases. In brief, we start the $2$-dimensional case as follows:
\begin{align} \frac{\text{d}}{\text{d}t} \frac{\partial x}{\partial x_0} & = \frac{\text{d}}{\text{d}t} \text{det} \large\begin{bmatrix} \frac{\partial x_1}{\partial x_{0,1}} & \frac{\partial x_1}{\partial x_{0,2}} \\ \frac{\partial x_2}{\partial x_{0,1}} & \frac{\partial x_2}{\partial x_{0,2}} \end{bmatrix}, \\ & = \frac{\text{d}}{\text{d}t} \Big{(} \frac{\partial x_1}{\partial x_{0,1}} \frac{\partial x_2}{\partial x_{0,2}} - \frac{\partial x_1}{\partial x_{0,2}} \frac{\partial x_2}{\partial x_{0,1}} \Big{)} \tag{1} \label{1} \end{align}
We use the fact that $$\frac{\text{d}}{\text{d}t} x_1 = v_1,$$ and use the chain rule to take the Euclidean derivative of the velocity as follows:
$$\frac{\partial v_1}{\partial x_{0,1}} = \frac{\partial v_1}{\partial x_{1}} \frac{\partial x_1}{\partial x_{0,1}} + \frac{\partial v_1}{\partial x_{2}} \frac{\partial x_2}{\partial x_{0,1}}$$
We then rewrite \eqref{1} as
\begin{align} & \frac{\partial v_1}{\partial x_{1}} \Big{(} \frac{\partial x_1}{\partial x_{0,1}} \frac{\partial x_2}{\partial x_{0,2}} - \frac{\partial x_1}{\partial x_{0,2}} \frac{\partial x_2}{\partial x_{0,1}} \Big{)} \\ & + \frac{\partial v_1}{\partial x_{2}} \Big{(} \frac{\partial x_1}{\partial x_{0,1}} \frac{\partial x_2}{\partial x_{0,2}} - \frac{\partial x_1}{\partial x_{0,2}} \frac{\partial x_2}{\partial x_{0,1}} \Big{)} \\ & + ... \\ & = \frac{\partial x}{\partial x_0} \nabla \cdot v, \end{align}
and we are done for the $2$-dimensional case. The $3$-dimensional case can similarly be calculated explicitly, taking the correct determinants.
My plan for the general $n$-dimensional case was to use a Proof by Induction. I had hoped that we could assume the formula holds in an $(n-1)$-dimensional case, and then add an $n$-th term to all the vectors involved. I then tried to use a general method of taking determinants of $n\times n$-matrices in terms of smaller $(n-1) \times (n-1)$ determinants, but I have gotten completely lost in those calculations, and suspect that this is not how we are intended to solve the problem.
Any hints whatsoever are appreciated. Thank you.