I've been trying to study the lie algebra of derivations on a given associative, commutative algebra over $\mathbb{R}$ or $\mathbb{C}$. The motivation for this is to study properties of a smooth manifold through studying the derivations on its algebra of smooth functions. In reviewing literature on derivations, I have found many papers on local derivations. For example, this professor has local derivations as one of their research interests, and has dozens of papers on the subject.
For some background, a derivation is an $F$-linear operator on an $F$-algebra $D$ that satisfies the product rule $D(a\times b)=D(a)\times b+a\times D(b)$. A local derivation $L$ is a generalization of the concept of a derivation (and thus a bit of a misnomer, since it might not actually be a derivation) is an $F$-linear operator such that at each $a$ in the algebra, there is a derivation $D_a$ depending on $a$ that satisfies $L(a)=D_a(a)$. The main question in studying local derivations seems to be when they are derivations.
My question is: Why do we care about local derivations? Would they be connected to my research question of connecting the analytic properties of smooth manifolds with the algebraic properties of its lie algebra of derivations?