The basic idea of this question is to see if there is any other derivations than 'formal derivations'.


Let $\mathbb{K}$ be a field. Given a commutative $\mathbb{K}$-algebra $A$, a derivation of $A$ is a $\mathbb{K}$-linear map $D:A\rightarrow A$ satisfying $D(ab)=D(a)b+ aD(b)$. Consider the case when $\mathbb{K}=\mathbb{R}$ and the algebra $A$ is the formal power series $\mathbb{R}[[x]]$. One derivation of $\mathbb{R}[[x]]$ is $p(x)\frac{\partial}{\partial x}$, defined formally as
$$ p(x) \frac{\partial}{\partial x} \left(\sum\_{n=0}^\infty c\_n x^n \right)=p(x)\cdot \left(\sum\_{n=0}^\infty n\cdot c\_n x^{n-1}\right)$$
for any $p(x)\in \mathbb{R}[[x]]$. My question is that, are there any derivations that is not of the form $p(x)\frac{\partial}{\partial x}$ as above? If yes, is there any reference/proof/example for existence of such derivations?


This question arose when I was thinking of vector fields of a $\mathbb{Z}$-graded manifold as a derivation of smooth functions. With my notion of graded manifolds, smooth functions are formal power series of virtual coordinates.


When I was thinking about it, since $\mathbb{R}[[x]]$ is not generated by $\{x\}$ as an algebra, there should be some other derivations but I could not make it clear. Any help to figure it out would be greatly appreciated.



Derivation of formal power series
Every $\mathbb{K}$-derivation $D$ of $\mathbb{K}[[x\_1,\dots,x\_n]]$ has the standard form $$D(f)=\sum\_{i=1}^n p\_i \frac{\partial f}{\partial x\_i},$$ where of course $p\_i=D(x\_i)$.  

Indeed, let $\mathfrak{m}$ be the maximal ideal of $\mathbb{K}[[x\_1,\dots,x\_n]]$: then clearly $D(\mathfrak{m}^{N+1})\subset \mathfrak{m}^N$ for all $N\geq0$, so $D$ is continuous for the $\mathfrak{m}$-adic topology. The claim follows because the formula holds when $f$ is a polynomial, $\mathbb{K}[x\_1,\dots,x\_n]$ is dense in $\mathbb{K}[[x\_1,\dots,x\_n]]$, and $\mathbb{K}[[x\_1,\dots,x\_n]]$ is Hausdorff.


