Let $k$ be an algebraically closed field of characteristic $0$. Consider the local ring $S=k[x_1,...,x_d]_{(x_1,...,x_d)}$, and call its maximal ideal $\mathfrak m$. Let $f\in k[x_1,...,x_d]$ be such that $f\in \mathfrak m^2\setminus \mathfrak m^3$. Let $R=S/fS$. I believe that the module of Kahler differentials $\Omega_R$ fits into an exact sequence $R \xrightarrow{\Theta} R^{\oplus d }\to \Omega_R \to 0$, where theta is multiplication by the Jacobian matrix of $f$.
My question is: Is the map $\Theta$ injective?