Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions.
$(1)$ It is an endomorphism of the additive abelian group of $R;$
$(2)$ For any $r,s\in R$ we have that $d(rs)=d(r)s+rd(s).$
I believe that when $R$ is an $A$-algebra for some commutative ring $A,$ we should also make the assumption that $d$ respects scalar multiplication, but I haven't seen the definition other than in the case of $R=K[x]$ and $A=K$ for a field $K.$
In this case, more specifically for $K=\mathbb C,$ it is standard knowledge that the standard derivative satisfies these conditions. Actually, we could just as well take the algebra of all holomorphic functions (instead of polynomials only) with the standard derivative.
However, there are more such functions. It was a homework assignment in my field theory class to prove that for any $f\in K[x]$ there is exactly one derivation $d_f:K[x]\longrightarrow K[x]$ such that $d_f(x)=f.$ This is an easy exercise and one easily sees that in particular, for $K=\mathbb C,$ the standard derivative is just $d_1.$
I understand that I will soon be shown how such derivations can be used in algebra, and I can't wait to see that. But since it's a field theory class, I'm quite sure that nothing will be said about their meaning outside algebra. This is why I'm writing this.
I would like to know
$(a)$ whether the derivations $d_f$ on $K[x]$ I wrote about above have any interpretation connected to actually differentiating something (which I understand involves taking limits in some metric space) and what the interpretation is. In particular, can we meaningfully assign a metric space to an arbitrary field?
$(b)$ whether the general definition of a derivation on a ring or algebra has such an an interpretation.