Let $V\subset \mathbb{A}^n$ be a variety given by a single equation. Prove that a point $P\in V$ is nonsingular if and only if $$\text{dim}_{\bar{K}}M_P/M_P^2=\text{dim}V.$$
For a general variety $V$, this equality holds. I am trying to prove this particular case.
Thank you!
$\text{dim}V$: the transcendence degree of $\bar{K}(V)$ over $\bar{K}$
$M_P=\{f\in \bar{K}[V]:f(P)=0\}$ (a maximal ideal of $\bar{K}[V]$)
$P\in V$ is nonsingular if the matrix $$\bigg(\frac{\partial f}{\partial X_1}(P)\dots\frac{\partial f}{\partial X_n}(P)\bigg)$$ has rank $1$.