\[ \newcommand{\tr}{\Rightarrow} \newcommand{\trs}{\tr^{\!\ast}} \newcommand{\rlnm}[1]{\mathsf{(#1)}} \newcommand{\rred}[1]{\xrightarrow{#1}} \newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}} \newcommand{\cl}{\mathsf{Cl}} \newcommand{\pow}{\mathcal{P}} \newcommand{\matches}{\mathrel{\mathsf{matches}}} \newcommand{\kw}[1]{\mathsf{#1}} \]

Brzozowski Derivatives

\[\begin{array}{rcll} a^{-1}\,a &=& \epsilon &\\ a^{-1}\,b &=& \emptyset & \text{when $a \neq b$}\\ a^{-1}\,\epsilon &=& \emptyset & \\ a^{-1}\,\emptyset &=& \emptyset & \\ a^{-1}\,(R \cdot S) &=& a^{-1}\,R \cdot S & \text{when $R$ is not nullable}\\ a^{-1}\,(R \cdot S) &=& a^{-1}\,R \cdot S + a^{-1}\,S & \text{when $R$ is nullable}\\ a^{-1}\,(R + S) &=& a^{-1}\,R + a^{-1}\,S &\\ a^{-1}\,(R^*) &=& a^{-1}\,R \cdot R^* & \end{array}\]