Let $M$ be the graded algebra of modular forms for $\operatorname{SL}_2(\Bbb Z)$. It is generated by $Q = E_4, R= E_6$, the Eisenstein series of weight $4$ and $6$ respectively. If we define: $$P = E_2 = 1 - 24\sum_{n\geq 1}\sigma_1(n)q^n$$ and the operator on $M$: $$\theta(f(q)) = q\frac{df}{dq} = \frac{1}{2\pi i}\frac{df}{dz},$$ then we can define a derivation on $M$ by: $$Df = 12\theta f - kPf$$ where $f$ is a modular form of weight $k$. I have a few questions about this operator:
Question 1: I have heard it said that this is a derivation on $M$. Does this mean that $D(fg) = D(f)g + fD(g)?$ I think this is clear for $\theta$ but it does not seem to be true for $P$. Am I making a mistake here or does derivation mean something else?
Question 2: What is the conceptual reason that one would expect there to be a such a derivation on the space of modular forms? What happens for other congruence subgroups?
Question 2-b What is the conceptual significance of $P$ behaving almost like a modular form?