Left and right acting derivatives

Saw an interesting notation in this Math Stackexchange answer . I quote - “ Eq. $(1)$ may be rewritten as $𝐷(𝑓\cdot 𝑔)=𝑓(\stackrel{⃖}{\text{D}}+\vec{\text{D}})𝑔$, where $\stackrel{⃖}{\text{D}}$ acts on...