In my lecture we defined:
For $(a,b)=P\in V(f)$, $P$ is a singular point if $\operatorname{mult}(f,P)>1$, where $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, and $\mathfrak{m}_{P}=(x-a,y-b)$. ($f\in k[X,Y]\setminus k$, $P\in\mathbb{A}^2$).
Now in Shafarevich's 'Basic Algebraic Geometry' he defined:
Let $C=V(f)$ be a curve. $P$ is a singular point of $C$, if $f_x '(P)=f_y'(P)=f(P)=0$.
I don't see how these two definitions are the same. It would be really nice if someone could explain me how these two definitions are connected. Thanks and best, Luca!