Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then the curve is singular at this point. My question is: do the partials tell us what kind of singularity there is? That is, would we be able to detect a cusp, node, etc. just by looking at the partials?
More generally, if we have some space curve, $\mathcal{C} \subset \mathbb{A}^n_{\mathbb{C}}$, the minors of the Jacobian cut out the singular locus. Can they tell us what kind of singularity we have?