Hensel's Lemma/P-adic Integers/Lemma 8

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\map F X \in \Z_p \sqbrk X$ be a polynomial.

Let $\map {F'} X$ be the (formal) derivative of $F$.

Let $k \in \N_{>0}$.

Then:
 * $x, y \in \Z_p \implies \map F {x + y p ^k} \equiv \map F x + y p^k \map {F'} x \pmod {p^{k+1}\Z_p}$

Proof
Let $\map F X = \ds \sum_{j \mathop = 0}^r c_j X^j$ where $X$ is the indeterminate and $c_0, c_1, \ldots, c_r \in \Z_p$.

Then:
 * $\map {F'} X = \ds \sum_{j \mathop = 1}^r j c_j X^j$

We have: