Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).

In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is an ideal of functions vanishing at $x$ in ring $C^\infty(M)$ (or $C_x^\infty(M)$). One can prove this isomorphism algebraically, or using Hadamard's lemma like argument.

In AG, given an affine variety $V$ and $x\in V$ one defines Zariski cotangent space as $\mathfrak{m}_x/\mathfrak{m}_x^2$, where $\mathfrak{m}_x$ is defined as an ideal of functions vanishing at $x$ in ring $O_{V,x}$.

Clearly AG introduced its version of tangent and cotangent spaces based on DG. However, It seems to me that in DG the description of cotangent space as $I_x/I_x^2$ is rarely used.

Question.Was the description of the cotangent space $T^*_xM$ as $I_x/I_x^2$ been known before cotangent Zariski space was introduced?