Let $k$ be a field, let $F \in k[X,Y,Z]$ be a nonzero homogeneous polynomial, and let $C \subset \mathbb P^2_k$ be the plane curve $F=0$. A point $P \in C(\overline{k})$ is nonsingular if and only if at least one of the partial derivatives $F_X(P)$, $F_Y(P)$, and $F_Z(P)$ is nonzero, equivalently $(F_X(P),F_Y(P),F_Z(P))\neq(0,0,0)$.