Let $Y = X \backslash Z(f)$ be a quasi-affine variety for some affine variety $X$, and let $p \in Y$. I'd like to prove that $T_p Y \cong T_p X$.
I have the following definition of $T_p X$:
If $A$ is a $k$-algebra and $\phi : A \to k$ a $k$-algebra homomorphism, then a $k$-linear map $D : A \to k$ is a 'derivation centred at $\phi$' if $D(fg) = \phi(f)D(g) + D(f)\phi(g)$ for all $f,g \in A$. Let $\mathrm{Der}(A,\phi)$ be the vector space of derivations centred at $\phi$. Define $T_p X = \mathrm{Der}(k[X], \mathrm{ev}_p)$.
My thoughts so far:
A morphism $\psi : Y \to X$ of affine varieties gives rise to a $k$-algebra homomorphism $\psi^*: k[X] \to k[Y]$ which maps $g \mapsto g \circ \psi$. This in turn gives rise to a $k$-linear map $d_\psi : T_pY \to T_{\psi(p)}X$, by sending $D \mapsto D \circ \psi^*$.
So for the case in question, we have a morphism $i : Y \to X$ given by the inclusion map, under which $i(p) = p$, so this looks pretty promising. I need to show that the map $d_i$ is a bijection.
If $d_i(D) = 0$, then $D \circ i^* = 0$ - I think this can only happen if $D$ is the zero map, so we have injectivity. Surjectivity I'm unsure about, and I haven't used $f$ anywhere yet. Any hints would be appreciated
Thanks