What's an example of algebra where differential operators aren't generated by order 1?

Question

For any commutative algebra $A$ over a field $k$, one can define its algebra of differential operators $\def\Diff{\operatorname{Diff}} \Diff_*(A)$, which has a filtration by order. In many cases the whole $\Diff_*(A)$ is generated by $\Diff_1(A)$, but not always. Can you please tell me an example of $A$ for which it's not the case?

Answer 1

Okay, I got it myself. Let $p=\operatorname{char}K$. Then $\mathrm{Diff}_*K[x]$ is not generated by operators of order one. Indeed, consider an operator $D^{[p]}$ which is defined by $D^{[p]}(x^n)=C^p_nx^{n-p}$. It's a differential operator, as in zero characteristic it's just $\frac{1}{p!}(\frac{d}{dx})^p$, and it clearly can't be expressed through $x$ and $\frac{d}{dx}$ in characteristic $p$.