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A Noetherian scheme $X$ over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}_{X,x}$.

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A possible reference is

A. Borel, Linear Algebraic Groups, AG.15.4, p.34.

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