Inconsistencies in the definition of derivative of a polynomial over a field A problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n na_nx^{i-1}$." However, there seem to be some problems with this definition. First, we cannot assume that $F$ contains the natural numbers as a subset. For example, what is $D(x^2)$ if $F = \mathbb{Z_2}$? Furthermore, the derivative may not be well-defined, depending on the definition of equality for polynomials. For example, if $F = \mathbb{Z_3}$, then $2x^2 + x = 0$ for $x \in \{0, 1, 2\}$, but $D(2x^2 + x) = 2(2)x+1 = x + 1 \neq 0 = D(0)$.
The problem then defines the derivative of rational functions $P/Q$ with the standard quotient rule formula $D(P/Q) = \frac{P'Q - PQ'}{Q^2}$ and then asks to prove that that definition is well-defined, but I can't do that without knowing that the derivative of polynomials exists and is well-defined. Can anyone shed some light on how to interpret the definition given, or what restrictions need to be made to make it work?

*

*First Issues


If $F$ is a field, we always identify the natural number $n$ with the element 
$$1+1+..+1 \in F$$
This definition can be extended easily to $\mathbb Z$. 
So, to answer the question
$$D(x^2)=(1+1)x=x+x$$


*

*Second issue


First, $2x^2+x \neq 0$ when $x=2$.
Also, you are making a very common mistake here. Even if the corresponding functions would be the same, the polynomials $2x^2+x$ and $0$ would still be different polynomials over $\mathbb Z_3$, and hence their derivatives would not need be equal.
A polynomial is just an algebraic extension. Over infinite fields, polynomials correspond uniquely to a function, but over finite fields you cannot identify polynomials with their corresponding function, you have to think about them just as polynomials.
