Derivation linear map examples From Humphreys' Introduction to Lie Algebras and Representation Theory:

By an $F$-algebra (not necessarily associative) we simply mean a vector space $U$ over $F$ endowed with a bilinear operation $U\times U\rightarrow U$, usually denoted by juxtaposition (unless $U$ is a Lie algebra, in which case we always use the bracket). By a derivation of $U$ we mean a linear map $\delta:U\rightarrow U$ satisfying the familiar product rule $\delta(ab)=a\delta(b)+\delta(a)b$.

I'm wondering what is an example of a derivation. Suppose I take $U=\mathbb{R}^n$. Clearly the map that takes everything to $0$ is a derivation. The map $\delta(x)=kx$ is not a derivation for $k\neq 0$, because then $kab\neq kab+kab$. What are some other linear maps satisfying that product rule?
 A: As I commented above, I'm not sure what $\mathbb{R}$-algebra structure you're referring to on $\mathbb{R}^n$. The cross product endows $\mathbb{R}^{3}$ with the structure of an $\mathbb{R}$-algebra but this isn't applicable in higher dimensions. You can, however, consider the quaternions, the octonions etc. as these furnish $\mathbb{R}$-algebras. The following answer provides some examples of derivations and furnishes some practice with the notion.
The polynomial ring $\mathbb{R}[x]$ is an $\mathbb{R}$-algebra. A derivation $\delta:\mathbb{R}[x]\to \mathbb{R}[x]$ is given by the rule $\delta(f)=\frac{df}{dx}$. (In fact, this example is the initial motivation for the definition of "derivation".) The set of derivations of an $\mathbb{F}$-algebra is an $F$-vector space. In particular, any scalar multiple of $\delta$ is again a derivation in the example of the first two sentences of this paragraph.
In general, a derivation $\delta$ of an $F$-algebra $U$ has the following properties:
(1) $\delta(1)=0$ if $1$ is the unity of $U$.
(2) $\delta(u^n)=n\delta(u^{n-1})$ for all $u\in U$.
Exercise 1: Prove (1) and (2).
Exercise 2: Classify all derivations of the $\mathbb{R}$-algebra $\mathbb{R}[x]$. (Hint: A derivation $\delta$ of $\mathbb{R}[x]$ is determined by $\delta(x)$. Why? Also, use Exercise 1.)
I hope this helps and provides some practice with the concept of a derivation! 
