Derivative of a Polynomial Function on the Real Line

Let $\mathbb{R}$ be the real numbers and let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ as in that definition. Let $\iota_{\mathbb{R}}:\mathbb{N}\to\mathbb{R}$ be the canonical map of $\mathbb{R}$, as in clause 3 of The Real Numbers and Standard Notation (written explicitly here, rather than by the abbreviation of that clause, because natural numbers occur below both as exponents and as elements of $\mathbb{R}$), and let powers be the natural number powers of $\mathbb{R}$. For a map $f:\mathbb{R}\to\mathbb{R}$ and a subset $I\subseteq\mathbb{R}$ write $f|_{I}$ for the restriction of $f$ to $I$; differentiability is that of Derivative at an Interior Point.

Then the following hold.

1. (Powers) Let $I\subseteq\mathbb{R}$ be an interval and let $x_{0}\in I$ be an interior point of $I$. Then the restriction to $I$ of the map $x\mapsto x^{1}$ is differentiable at $x_{0}$ with derivative $1$; and for every $m\in\mathbb{N}$ the restriction to $I$ of the map $x\mapsto x^{S(m)}$ is differentiable at $x_{0}$ with derivative $\iota_{\mathbb{R}}(S(m))\,x_{0}^{m}$.

2. (Polynomial functions) For every polynomial function $p$ on $\mathbb{R}$ there is a polynomial function $p^{\ast}$ on $\mathbb{R}$, depending only on $p$, such that for every interval $I\subseteq\mathbb{R}$ and every interior point $x_{0}$ of $I$ the restriction $p|_{I}$ is differentiable at $x_{0}$ with derivative $p^{\ast}(x_{0})$.