 Multiplication of a distribution by a smooth function

For a distribution $u$ and a smooth function $a$, the product $au$ is defined by $⟨au,\phi ⟩=⟨u,a\phi ⟩$. It obeys the Leibniz rule; in one dimension, $(au{)}^{\prime }=a^{\prime }u+a{u}^{\prime }$.