Section 7.2 of the notes on etale cohomology (https://people.dm.unipi.it/tdn/CoomologiaEtale/Note.pdf) by Lombardo and Maffei states the following (up to a small typo):
Let $k$ be any field, let $A = \frac{k[x_1,...,x_n]}{\langle f_1,...,f_m \rangle}$, and let $Jf = [\frac{\partial f_j}{\partial x_i}]$ be the Jacobian matrix of the $f_j$. Let $\mathfrak{m}$ be a maximal ideal of $A$, and $k(\mathfrak{m}) = \frac{A}{\mathfrak{m}}$. Then $\dim_{k(\mathfrak{m})} \frac{\mathfrak{m}}{\mathfrak{m}^2} \le n - $ rank $Jf(\mathfrak{m})$, where $Jf(\mathfrak{m})$ is obtained from $Jf$ by reducing all entries mod $\mathfrak{m}$.
The proof is omitted in the notes. Can anyone please supply a proof or a reference?