Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\setminus\{0\}$.

Is it true that $\mathbb{C}[x_1,\ldots, x_n] = \mathbb{C}[f_1,\ldots, f_n]+f_n\cdot\mathbb{C}[x_1,\ldots, x_n]$?