Suppose $\varphi = (\varphi_2, \ldots, \varphi_n) : \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ is a $C^2$ function, i.e. all second order partial derivatives of the $\varphi_i$ exist and are continuous. Show that the symbolic determinant
$$\left| \begin{array}{cccc}
\frac{\partial}{\partial x_1} & \frac{\partial \varphi_2}{\partial x_1} & \ldots & \frac{\partial \varphi_n}{\partial x_1} \\
\vdots & \vdots & & \vdots \\
\frac{\partial}{\partial x_n} & \frac{\partial \varphi_2}{\partial x_n} & \ldots & \frac{\partial \varphi_n}{\partial x_n}
\end{array} \right|$$
vanishes identically.