For $X$ an affine variety and $p \in X$ define $T_p X = \mathrm{Der}(k[X], \mathrm{ev}_p)$.
Claim: If $Y = X \setminus Z(f)$ is some Zariski open affine subvariety of $X$ and $p \in Y$, then $T_p X \cong T_pY$.
Definition: Let $X$ be an arbitrary variety, $p \in X$. Define $T_p X = T_p U$ for any open affine neighbourhood $U$ of $p$.
Apparently the above claim tells us this definition makes sense, but I can't see why. I'd appreciate an explanation (note that I'm only really concerned with projective varieties, so if a simpler explanation can be afforded by restricting to this case then please do so!)
Thanks