The derivative of a distribution \(u\in \mathcal{D}(\Omega)\) into the direction \(e_k\) is defined by

\begin{equation*} (\partial_k u)(\varphi)=-u(\partial_k\varphi), \end{equation*}

where \(\varphi\in \mathcal{D}(\Omega)\). This defines a new distribution.

Remarks
  • The classical derivative coincide with the distributional one. This can be seen by applying integration by parts.