First of all I'll motivate my question: when we define the tangent space to a smooth manifold $M$ we can go through two different approaches that at the end give the same result.
First approach We pick $U \subset X$ an open subset and $p \in U$. We define a derivation at $p$ as a $\mathbb{R}$ linear map $D: C^{\infty}_X(U) \rightarrow \mathbb{R}$ that verifies Liebntiz's condition at $p$, i.e. $D(f \, g) = D(f) \, g(p) + f(p) \, D(g)$. We then define the tangent space in $p$ at $U$ as $T_{p, U} = \{ D: C^{\infty}_X(U) \rightarrow \mathbb{R} \; \text{derivation at $p$}\}$. Using bump functions one can prove that the inclusion $i : U \hookrightarrow X$ induces an isomorphism between the tangent spaces at $p \in U$ and so the concept of tangent space turns out to be local.
Second approach We do the same except that we define a derivation as a $\mathbb{R}$ linear map $D: C^{\infty}_{p,X} \rightarrow \mathbb{R}$, where with $C^{\infty}_{p,X}$ it's the stalk at $p$ of the sheaf $C^{\infty}_X$. In this way the tangent space is defined as a local concept.
My question deal with the holomorphic tangent space. It is defined via derivation at a point $p$: $\mathbb{C}$ linear map $D: \mathcal{O}_{p,X} \rightarrow \mathbb{C}$. Is the reason for which it is defined this way that we don't have a holomorphic analogous of $C^{\infty}$ partitions of unity? I guess it is, but I would like a confirmation.
If we had defined the holomorphic tangent space as we did in the first approach for the smooth tangent space, what would have been the problem? Would have we had some pathological situation?
Thank you!