Localization of Kähler differentials (${\Omega }_{S^{-1}A/k}=S^{-1}{\Omega }_{A/k}$)

Kähler differentials commute with localization of a ring. The quotient rule $d(a/s)=(sda-ads)/s^2$ uniquely extends a derivation after inverting $s$; its universal property proves the displayed module isomorphism.