My understanding of how derivations on commutative rings are like derivatives is that a derivation on $ R$ is differentiation with respect to a vector field on $ \text{Spec}(R)$ . But derivations are supposed to be thought of as like derivatives in a wider context than commutative rings, and I don’t really understand how.

Take anti-derivations on the exterior algebra of differential forms on a manifold, for instance. The exterior derivative and Lie derivatives both give you information about infinitesimal change in a differential form, but the interior derivative is defined pointwise, as an anti-derivation on the exterior algebra of the tangent space at each point, which ruins any attempt to think of anti-derivations on differential forms as capturing some information about infinitesimal change. So how can you think about interior differentiation as being like a derivative in any more concrete sense than that it obeys similar syntactic rules? More generally, how can you think about anti-derivations on exterior algebras (or more generally still, on anti-commutative graded rings) as being like a derivative?

There’s also derivations on non-commutative rings. The adjoint action of an element of a ring $ \text{ad}_x(y):=xy-yx$ is a derivation, but I don’t see the significance of this. For example, the Pincherle derivative seems to act like a sort of “differentiation with respect to $ d/dx$ ” insofar as it sends $ d/dx$ to $ 1$ , and the fact that it is a derivation forces certain other facts that this heutristic naively suggests to be true (for instance, that the shift operator $ S_1=e^{d/dx}$ is its own Pincherle derivative). Is there some more precise way to describe the Pincherle derivative as differentiation with respect to $ d/dx$ ? What about a way to characterize arbitrary derivations on non-commutative rings?

How about derivations on Lie algebras? The Jacobi identity can be interpreted as saying that adjoint actions are derivations, but as in the analogous fact I mentioned for derivations on non-commutative rings, I’m curious about what the significance of this is. And about how to think of arbitrary derivations on Lie algebras.