We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define $$ D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar{a}b, a\otimes a^{\prime}=aa^{\prime}\otimes 1). $$ And Theorem 19.4.1 of Ginzburg's notes claims that: An associative algebra $A$ is formally smooth if and only if the natural map $A\to \bar{A}$ can be extended to a derivation of $D(A)$ of degree $+1$.

Ginzburg also points out that the commutative polynomial ring $k[x_1\ldots x_n]$ is not formally smooth for any $n>1$.

My question is why $k[x,y]$ is not formally smooth in the sense of the above theorem, i.e, what is the difficulty to extend the morphism $x\to \bar{x}, y\to \bar{y}$ to a derivation on $D(k[x,y])$?