Let $k$ be a field, let $X$ be an affine variety over $k$, and let $A(X)$ be its coordinate $k$-algebra with unit $1_{A(X)}$. Let $p\in X(k)$ be a $k$-rational point, and let $\operatorname{ev}_p:A(X)\to k$ be the evaluation $k$-algebra homomorphism at $p$. Set $\mathfrak m_p:=\ker(\operatorname{ev}_p)=\{f\in A(X):f(p)=0\}$. If the Zariski tangent space at $p$ is defined by $T_pX:=\operatorname{Hom}_k(\mathfrak m_p/\mathfrak m_p^2,k)$, then there is a natural isomorphism of $k$-vector spaces between $T_pX$ and the space of $k$-linear maps $D:A(X)\to k$ satisfying $D(fg)=f(p)D(g)+g(p)D(f)$ for all $f,g\in A(X)$.