Proof of Derivative of a Polynomial Function on the Real Line

Write $\iota=\iota_{\mathbb{R}}$. Throughout, $I$ denotes an interval in $\mathbb{R}$ and $x_{0}$ an interior point of $I$.

Step 1 (the first power). By claim 1 of Properties of Natural Number Powers in a Field we have $x^{1}=x$ for every $x\in\mathbb{R}$, so the restriction to $I$ of $x\mapsto x^{1}$ is the map $z\mapsto z$ on $I$. Let $\varepsilon>0$ and take $\delta=1$. If $h\in\mathbb{R}$ satisfies $0<|h|<\delta$ and $x_{0}+h\in I$, then

so the quantity $|1-1|=|0|=0$ is smaller than $\varepsilon$. By Derivative at an Interior Point the restriction is differentiable at $x_{0}$ with derivative $1$.

Step 2 (claim 1). Let $E$ be the set of those $m\in\mathbb{N}$ with the following property: for every interval $I$ and every interior point $x_{0}$ of $I$, the restriction to $I$ of $x\mapsto x^{S(m)}$ is differentiable at $x_{0}$ with derivative $\iota(S(m))\,x_{0}^{m}$.

Base. By claim 1 of Properties of Natural Number Powers in a Field, $x^{S(1)}=x^{1}x=xx$, so the restriction to $I$ of $x\mapsto x^{S(1)}$ is the pointwise product of the map $z\mapsto z$ with itself. By Step 1 and claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives it is differentiable at $x_{0}$ with derivative $1\cdot x_{0}+x_{0}\cdot 1=x_{0}+x_{0}$. On the other hand $S(1)=1+1$ by claim 1 of Arithmetic of Addition on the Natural Numbers, so $\iota(S(1))=\iota(1)+1=1+1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; since $x_{0}^{1}=x_{0}$, distributivity gives $\iota(S(1))\,x_{0}^{1}=(1+1)x_{0}=x_{0}+x_{0}$. Hence $1\in E$.

Step. Let $m\in E$. By claim 1 of Properties of Natural Number Powers in a Field, $x^{S(S(m))}=x^{S(m)}x$, so the restriction to $I$ of $x\mapsto x^{S(S(m))}$ is the pointwise product of the restrictions of $x\mapsto x^{S(m)}$ and $z\mapsto z$. By $m\in E$, Step 1 and claim 3 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives it is differentiable at $x_{0}$ with derivative

using claim 1 of Properties of Natural Number Powers in a Field for $x_{0}^{m}x_{0}=x_{0}^{S(m)}$ and distributivity. Finally $S(m)+1=S(S(m))$ by claim 1 of Arithmetic of Addition on the Natural Numbers, so $\iota(S(m))+1=\iota(S(S(m)))$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Hence $S(m)\in E$.

By Principle of Induction for the Natural Numbers, $E=\mathbb{N}$, which together with Step 1 proves claim 1.

Step 3 (claim 2). Let $E'$ be the set of those $N\in\mathbb{N}$ with the following property: for every $c_{0}\in\mathbb{R}$ and every map $c:[N]\to\mathbb{R}$ on the initial segment determined by $N$, the polynomial function $p$ given by $p(x)=c_{0}+\sum_{k=1}^{N}c_{k}x^{k}$, with the finite sum of $\mathbb{R}$, admits a polynomial function $p^{\ast}$ as in claim 2.

Base. For $N=1$, claim 1 of Properties of Finite Sums and claim 1 of Properties of Natural Number Powers in a Field give $p(x)=c_{0}+c_{1}x$. Restricted to $I$, $p$ is the pointwise sum of the constant function with value $c_{0}$ and the scalar multiple by $c_{1}$ of the map $z\mapsto z$; by claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives and Step 1 it is differentiable at $x_{0}$ with derivative $0+c_{1}\cdot 1=c_{1}$. The constant function $p^{\ast}$ with value $c_{1}$ is a polynomial function on $\mathbb{R}$ by claim 1 of Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions, and it does not depend on $I$ or on $x_{0}$. Hence $1\in E'$.

Step. Let $N\in E'$, let $c_{0}\in\mathbb{R}$ and let $c:[S(N)]\to\mathbb{R}$, and let $p(x)=c_{0}+\sum_{k=1}^{S(N)}c_{k}x^{k}$. Let $q(x)=c_{0}+\sum_{k=1}^{N}c_{k}x^{k}$, the sum being formed from the restriction of $c$ to $[N]$, which is legitimate by the restriction part of claim 1 of Properties of Finite Sums. By the recursion part of that claim and associativity of addition,

Since $N\in E'$ there is a polynomial function $q^{\ast}$ as in claim 2 for $q$. Let $w:\mathbb{R}\to\mathbb{R}$ be given by $w(x)=x^{N}$ and let $r:\mathbb{R}\to\mathbb{R}$ be given by $r(x)=c_{S(N)}x^{S(N)}$; both are polynomial functions on $\mathbb{R}$ by claims 1 and 2 of Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions. By claim 1, already proved, and claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, the restriction $r|_{I}$ is differentiable at $x_{0}$ with derivative $c_{S(N)}\,\iota(S(N))\,x_{0}^{N}$. Hence, by claim 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives again, $p|_{I}=q|_{I}+r|_{I}$ is differentiable at $x_{0}$ with derivative

The map $p^{\ast}=q^{\ast}+\bigl(c_{S(N)}\iota(S(N))\bigr)w$, a pointwise sum of a polynomial function and a scalar multiple of one, is a polynomial function on $\mathbb{R}$ by claim 2 of Constants, Powers, Sums, Scalar Multiples and Products of Polynomial Functions; it depends only on $p$, and the displayed derivative is $p^{\ast}(x_{0})$. Hence $S(N)\in E'$.

By Principle of Induction for the Natural Numbers, $E'=\mathbb{N}$. Since every polynomial function on $\mathbb{R}$ is of the form treated above for some $N\in\mathbb{N}$, by Polynomial Function on a Field, claim 2 follows.