Introduction
I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my mother tongue, sorry if I made grammar or vocabulary mistakes!
Tangent space to a point
Let $X$ be a projective variety ($X\subset \mathbb{P}^n$). The intrinsic definition of the tangent space to $X$ at $p\in X$ is $T_p(X)=(m_p/ m_p^2)^\star$, where $m_p$ is the maximum ideal of functions vanishing at p. Equivalently, one can define $T_p(X)= Z(\{d_pf ~|~ f\in I(X) \text{ homogeneous } \})$, where $Z(\{f_i\}_{i\in J\subset \mathbb{N}})=\{a\in \mathbb{P} | f_i(a)=0 \forall i\in J\}$.
Question
Interpretation of $T_p(X)=(m_p/ m_p^2)^\star$ : I don't understand the meaning of this definition. I understand, following $T_p(X)= Z(\{d_pf ~|~ f(t)=0 ~\forall t\in X\})$, that the tangent space to a point is the space where directional derivatives vanish, but I can't link this insight to the so-called intrinsic definition.
Edit 1 According to the reference in Qi Zhu's comment, here is what I have understood. Could you please tell me if I'm right ?
Derivatives around a point p are functions defined by two properties.
First, they must take something from $m_p$ in order to be polynomials and to be well defined at p, and send it to $\mathbb{R}$. Therefore they are first defined as functions $m_p \rightarrow \mathbb{R} : f \rightarrow f'(p)$.
Second, they must obey Leibniz rule, which expresses that if f(p)=g(p)=0, then (fg)'(p)=0. In other words, if f and g are polynomials and p=0, then (fg)'(p)=0. It means that the monomials terms of degree $\geq$2 of a polynomial won't change anything to the result of partial derivatives at such a point. Therefore, we can "drop" them, saying that two polynomials whose monomials terms of degree 1 are the same are equivalent. Then, derivatives at a point p are defined by functions $m_p/m_p^2 \rightarrow \mathbb{R} : f \rightarrow f'(p)$, that is, elements of $(m_p/m_p^2)^*$
I'm sure I was careless in the second part, but I'm not able to be more rigorous than that. If I'm right in the way to think it, I would appreciate and accept as an answer a rigorous correction of this explanation of what I have understood