Derivations in a ring. What applications do they have outside algebra? INTRODUCTION
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 $d_1.$
QUESTION
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.
 A: A derivation on an algebra $A$ is an infinitesimal automorphism, and the collection of derivations on $A$ naturally forms a Lie algebra which one should think of as the "Lie algebra of the automorphism group of $A$." When $A = C^{\infty}(\mathbb{R})$, the automorphism group corresponding to ordinary differentiation is translation $\alpha_t : f(x) \mapsto f(x+t)$, and more generally when $A = C^{\infty}(M)$ for $M$ a compact smooth manifold, a derivation on $A$ is the same thing as a vector field on $M$.
Not all derivations actually correspond to automorphisms. Given a derivation $D : A \to A$, formally the corresponding one-parameter family of automorphisms is supposed to be
$$e^{Dt} = \sum_{n \ge 0} D^n \frac{t^n}{n!}$$
and formally taking the derivative of this family recovers $D$, but the family usually only exists formally as a map $A \to A[[t]]$. However, if $A$ has extra structure then the above map may actually exist, for example if $A$ is a Banach algebra.
So derivations not only appear in algebra, but also in differential geometry and functional analysis. A longer explanation of what I just said above may be found at this blog post. 
A: Let $M$ be a differential manifold of class $\mathcal C^\infty$ and $m\in M$ be a point.  
1) A very elementary  and efficient way to define the tangent space $T_m(M)$ is to say that it is the space of derivations  of  the algebra of germs of smooth functions  defined in a neighbourhood of $m$:
$$T_m(M)=Der_{\mathbb R}(\mathcal C^\infty _m,\mathbb R)$$
2) In a similar vein the  space of global vector fields of $M$ can be identified with  the set of derivations of the algebra of global differentiable functions into itself: 
$$\mathcal X(M)=\Gamma(M,TM)=Der_{\mathbb R} (C^\infty (M),C^\infty (M) )$$   
3) In differential geometry there are alternative ways of defining the tangent space  $T_m(M)$: by equivalence classes of curves through $m$ or by giving a huge collection of vectors, one for  each chart  around $m$ , related by linear maps associated to each change of coordinates.
This last point of view  was popularized by   Einstein and other physicists more than a century ago.   
4) However in algebraic geometry, especially when studying singular varieties, derivations (or their variants like Zariski tangent space) are essentially the only tool for defining tangent spaces of  varieties.
For example, if you consider the cusp $M\subset  \mathbb A^2_k$ with equation $y^2-x^3=0$, its tangent space at the origin $m=(0,0)$ is the vector space $T_m(M)=k\cdot \frac {\partial} {\partial x} \oplus k\cdot \frac {\partial} {\partial y}$.
This is  weird since it is a $2$- dimensional vector space whereas  $M$ is $1$-dimensional.
This phenomenon, impossible for  differential manifolds, is due to  the singularity of $M$ at $m$.
