In Loring Tu's text An Introduction to Manifolds, Exercise $2.4$ asks us to show that $D_1\circ D_2$ need not be a derivation while $D_1\circ D_2-D_2\circ D_1$ is always a derivation. My question is the following: how should I interpret $D_1\circ D_2$?
For the background, a derivation is a map of real vector spaces $$D:C_p^\infty\to\mathbb{R}$$ that is $\mathbb{R}$-linear and satisfies the Liebniz rule: $$D(fg)=(Df)g(p)+f(p)(Dg),$$ where $p$ is some element in $\mathbb{R}$.
Clearly it doesn't make sense as a composition, since the domain and codomain are different, one not even being a subset of the other, so I'm inclined to think it is a product. However, if it is a product, then the second part of the question becomes trivial by the commutativity of $\mathbb{R}$, hence it is always zero.
Am I misunderstanding something? Please help!
