Let $X$ be an affine variety defined by a smooth curve $F=0$ on $\mathbb{A}^2$. Let $p=(a,b)$ be a point on $X$. Now $m_p$ be the maximal ideal generated by $x-a,y-b$. Then why $$F_X(p)(x-a)+F_Y(p)(y-b)\equiv 0\bmod{m_p^2}$$ Where $F_X$ is the partial derivative of $F$ with respect to $X$.
Now using this equation we obtain that $m_p/m_p^2$ has dimension 1 as a vector space of the residue field $k\cong \mathcal{O}_p/m_p$. Hence using nakayama's lemma we can conclude that $m_p$ is generated by one element.
But how $F_X(p)(x-a)+F_Y(p)(y-b)\equiv 0\bmod{m_p^2}$ is true.