\[
\newcommand{\tr}{\Rightarrow}
\newcommand{\trs}{\tr^{\!\ast}}
\newcommand{\rlnm}[1]{\mathsf{(#1)}}
\newcommand{\rred}[1]{\xrightarrow{#1}}
\newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}}
\newcommand{\cl}{\mathsf{Cl}}
\newcommand{\pow}{\mathcal{P}}
\newcommand{\matches}{\mathrel{\mathsf{matches}}}
\newcommand{\kw}[1]{\mathsf{#1}}
\]
Brzozowski Derivatives
\[\begin{array}{rcll}
a^{-1}\,a &=& \epsilon &\\
a^{-1}\,b &=& \emptyset & \text{when $a \neq b$}\\
a^{-1}\,\epsilon &=& \emptyset & \\
a^{-1}\,\emptyset &=& \emptyset & \\
a^{-1}\,(R \cdot S) &=& a^{-1}\,R \cdot S & \text{when $R$ is not nullable}\\
a^{-1}\,(R \cdot S) &=& a^{-1}\,R \cdot S + a^{-1}\,S & \text{when $R$ is nullable}\\
a^{-1}\,(R + S) &=& a^{-1}\,R + a^{-1}\,S &\\
a^{-1}\,(R^*) &=& a^{-1}\,R \cdot R^* &
\end{array}\]