Generalization of the derivative to polynomial rings It is easy to see why the derivative plays an important role in real and complex analysis from the geometric viewpoint. However, one can extend the definition of a derivative to polynomial rings such as $F_q[x] \text{ or } \Bbb Q[x]$ over the finite fields or rationals. 
We do this by simply defining $f'(x) = \sum_{k=1}^{n}ka_kx^{k-1}$ where $f(x) = \sum_{k=0}^{n}a_kx^k$ and they are still very useful, for instance a polynomial $f$ has multiple roots only if $f$ and $f'$ have a non trivial common divisor.
Is there some point of view from which there is actual meaning(geometric or otherwise) to the derivative? That is, if there was some mathematician who had never been exposed to analysis but did learn a lot of algebra, could we reasonably expect her to come up with the concept of a derivative?
 A: Let $R$ be any commutative ring. The derivative should be a function $d : R[x] \to R[x]$ with the following properties:


*

*$d(x)=1$

*$d$ is $R$-linear

*$d$ satisfies the Leibniz rule $d(f \cdot g)=f \cdot d(g) +  d(f) \cdot g$


One can show that $d$ is uniquely determined by these properties, and that it exists. Namely, one has $d(\sum_k u_k x^k)=\sum_k (k+1) u_{k+1} x^k$. A suitable generalization concerning derivations into arbitrary $R[x]$-modules shows that the module of Kähler differentials $\Omega^1_{R[x]/R}$ is free of rank $1$. This says that the tangent bundle of the affine line $\mathbb{A}^1_R$ is trivial. This is basically the beginning of a story which combines methods from differential geometry and algebraic geometry. In particular, Kähler differentials allow us to define smooth schemes, which are similar to smooth manifolds.
Here is another perspective: For $f \in R[x]$ consider $f(x+\varepsilon) \in R[x][\varepsilon]/(\varepsilon)^2$ in the ring of dual numbers over $R[x]$. Modulo $(\varepsilon)$ this is clearly $f(x)$. Hence, there is a unique polynomial $d(f)=f' \in R[x]$ such that $f(x+\varepsilon)=f(x) + \varepsilon \cdot f'(x)$. A more sloppy way of writing this is
$$f'(x)=\frac{f(x+\varepsilon)-f(x)}{\varepsilon}.$$
Now you can derive the properties of the derivative using this definition, which of course heavily reminds us of the definition of the derivative in analysis, and even more in non-standard analysis. The only difference is that the limit $\varepsilon \to 0$ is replaced by a specific ring element $\varepsilon$ satisfying $\varepsilon^2=0$.
Let me also briefly mention that the derivative also makes sense for formal power series. In fact, we may use the same characterizations as above. In the theory of combinatorical species, there is a combinatorial meaning of the derivative of formal power series. This makes it possible to solve combinatorial problems by means of differential equations.
Derivatives of polynomials appear frequently. But I don't know how to motivate them without a little bit of analysis. Maybe, generally speaking, it is a bad idea to blend certain areas of mathematics out. Rather, it is a good idea to merge some (all!) of them.
