Does the following elementary result have a name (or a reference to)?
Given a field $K$, and a polynomial $P(x) \in K[x]$, divide the polynomial $P(x) - P(y)$ by $(x - y)$ in $K[x][y]$: $P(x) - P(y) = D_P(x, y)\cdot (x - y) + R(x)$. Setting $y$ to $x$ obtains $R(x) = 0$.
Now, $\partial_x P(x) = D(x, x)$. Proof: directly check that $D_t = 0$, for $t \in K$, $D_x = 1$, $D_{P+Q} = D_P + D_Q$ and $D_{P\cdot Q} = P\cdot D_Q + Q\cdot D_P$.
This gives a nice "analysis-like" formula $P(x) = P(y) + D_P(x, y)\cdot(x - y)$.
When $K=\mathbb{R}$, this is Taylor's theorem. Perhaps it is more recognizable if we write it as $P(x+h) = P(x) + h\cdot Q(x,h)$. The fact that $Q$ is a polynomial follows from the fact that all but finitely many of the derivatives of $P$ are zero.
Note that since $Q$ is arbitrary, the only real content of the statement (even for general $K$) is that $h$ divides $P(x+h)-P(x)$.