Suppose $f = \left( f_1, f_2, \ldots, f_{n-1} \right) : \mathbb{R}^{n} \mapsto \mathbb{R}^{n-1}$ is a $C^{2}$ function, then show that the symbolic determinant
\begin{align} \begin{vmatrix} \frac{\partial}{\partial x_{1}} &\frac{\partial f_{1}}{\partial x_{1}} &\frac{\partial f_{2}}{\partial x_{1}} &\cdots &\frac{\partial f_{n-1}}{\partial x_{1}} \\\\ \frac{\partial}{\partial x_{2}} &\frac{\partial f_{1}}{\partial x_{2}} &\frac{\partial f_{2}}{\partial x_{2}} &\cdots &\frac{\partial f_{n-1}}{\partial x_{2}} \\ \vdots &\vdots &\vdots &\ddots &\vdots \\ \frac{\partial}{\partial x_{n}} &\frac{\partial f_{1}}{\partial x_{n}} &\frac{\partial f_{2}}{\partial x_{n}} &\cdots &\frac{\partial f_{n-1}}{\partial x_{n}} \end{vmatrix} \end{align}
vanishes identically.
I have been trying to rack my brains thinking of various methods which can be used to solve the following problem, but I am getting nowhere, I am not particularly good at theoretical multivariable calculus, and hence might be missing some basic concept here. I would be thankful if someone could point out a direction for me to work through.
P.S. This problem is from the entrance examination, 2010 to the Graduate School at Chennai Mathematical Institute.