Density of the smooth locus
= Density of the smooth locus
If $X\subseteq\mathbb A^n$ is irreducible of dimension $d$, choose a nonzero $(n-d)$-rowed minor of a Jacobian matrix at one point of minimum tangent dimension. Its nonvanishing locus is a nonempty <Zariski-open set> on which the Jacobian has rank $n-d$, so every point there is smooth. Irreducibility makes every nonempty <Zariski-open set> dense.