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this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes:

Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ($F,G\in\mathbb{F}[x_0,x_1,x_2]$ homogenous polynomials) and $P\in V(F)\cap V(G)$ a point. Let $\mu_P(F,G)$ denote the intersection number (multiplicity), i.e. the number, defined in Fultons book on p. 36, section 3.3 and also chapter 5. Is there any relationship between $\mu_P(F,G)$ and the possible equality of partial derivatives $$\frac{\partial F}{\partial x_i}\!\!\!(P)\overset{?}{=}\frac{\partial G}{\partial x_i}\!\!\!(P),\;\ldots,\;\frac{\partial^k F}{\partial x_i^k}\!\!\!(P)\overset{?}{=}\frac{\partial^k G}{\partial x_i^k}\!\!\!(P)\;?$$ Do the equalities hold for $k=\mu_P(F,G)$? If not, what is the correspondence? I assume degrees of $F$ and $G$ must be involved?

Thank you.

P.S. Some references from (preferably recent) books are highly desirable.

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