If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring $\mathbb{R}[x,y]$, it seems that $\frac{d}{dx}$ takes us into the "polynomial ring" $\mathbb{R}[x,y,\frac{dy}{dx}].$ We might write:
$$\frac{d}{dx} : \mathbb{R}[x,y] \rightarrow \mathbb{R}[x,y,\frac{dy}{dx}]$$
Perhaps this satisfies a universal property that explains why it feels like a natural thing to do. Anyway:
Question. Broadly speaking, how does commutative and/or differential algebra think about total derivatives, and from which book (article, etc.) can I learn this way of thinking?