In "Algebraic Geometry over the Complex Numbers", Arapura gives the following definition:
Definition 2.5.8. When $(R, m, k)$ is a local ring satisfying the tangent space conditions, we define its cotangent space as $T_R^* = m/m^2 = m \otimes_R k$, [...]
To give an example, he restricts to $\mathbb{R}^n$. Then $R$ is the ring of germs of $C^\infty$ functions at $x=0$ and the maximal ideal $m$ consists of those germs that vanish at $0$; it is generated by the coordinate functions $\{ x_i \}$. $k = R/m = \mathbb{R}$. Given a function $f$ with Taylor expansion $f(x) = f(0) + \sum \frac{\partial f}{\partial x_i}\vert_0 x_i + \mathcal{O}(x^2)$, the second term corresponds to the elements of $m/m^2$. What I do not understand is how that space is supposed to be isomorphic to the tensor product $m \otimes_R k$.Naively, $m$ includes the "second order behavior" that was discarded in the quotient, I don't see how tensoring with $k$ improves that.