Base change for Kähler differentials (${\Omega }_{(A{\otimes }_k{k}^{\prime })/k^{\prime }}={\Omega }_{A/k}{\otimes }_k{k}^{\prime }$)

The Kähler differentials commute with extension of the base ring. A derivation of the base-changed algebra that kills $k^{\prime }$ is determined by its restriction to $A\otimes 1$, and the Leibniz rule provides its unique extension. No flatness hypothesis is required for this isomorphism.