When a grammar quotient ${𝐷}_{𝐴}𝐵$ is taken with respect to a representable grammar $𝐴=\lceil 𝑤\rceil$ (matching exactly the string $𝑤$), it coincides with the residual $𝐴⊸𝐵$, as established by the Quotient and Residual Coincidence.

This special case is known as the Brzozowski derivative of $𝐵$ by the string $𝑤$, often written as ${𝜕}_{𝑤}𝐵$. We have the following coincidence: