Universal property of Kähler differentials (${\mathrm{Hom}}_A({\Omega }_{A/k},M)\cong {\mathrm{Der}}_k(A,M)$)

Composing an $A$-linear map with $d:A\to {\Omega }_{A/k}$ gives a $k$-derivation. Every derivation factors uniquely this way because its linearity and product rule are exactly the defining relations of the Module of Kähler differentials.