The chain rule applied to $x\mapsto (x^2+1{)}^5$ and to $x\mapsto ((3x-1{)}^2+2{)}^3$, with the Carathéodory factor written out in closed form in the first case

Numerals denote canonical naturals of $\mathbb{R}$ (The canonical natural $\iota (n)=n\cdot 1_F$ of a field) and powers are those of Integer powers $a^m$.

Claim 1. Let $F:\mathbb{R}\to \mathbb{R}$ be $F(x)=(x^2+1{)}^5$. Then $F$ is differentiable at every $c\in \mathbb{R}$ (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set) and

$F^{\mathrm{\prime }}(c)=\iota (10)c(c^2+1{)}^4\mathrm{.}$

Claim 2. For $b\in \mathbb{R}$ the Carathéodory factor (Carathéodory's characterisation: $f$ is differentiable at $c$ if and only if there is $\phi :A\to \mathbb{R}$, continuous at $c$, with $f(x)-f(c)=\phi (x)(x-c)$ for every $x\in A$, and then $\phi$ is unique and $\phi (c)=f^{\mathrm{\prime }}(c)$) of $y\mapsto y^5$ at $b$ is the polynomial function

$\phi (y)={\sum }_{k<5}{b}^k{y}^{4-k},$

which satisfies $y^5-b^5=\phi (y)(y-b)$ for every $y\in \mathbb{R}$, is continuous at $b$, and has $\phi (b)=\iota (5)b^4$.

Claim 3. Let $G:\mathbb{R}\to \mathbb{R}$ be $G(x)=((\iota (3)x-1{)}^2+2{)}^3$. Then $G$ is differentiable at every $c\in \mathbb{R}$ and

$G^{\mathrm{\prime }}(c)=\iota (18)(\iota (3)c-1)((\iota (3)c-1{)}^2+2{)}^2\mathrm{.}$

Claim 2 is included because it makes the mechanism of the chain rule visible: the factor that the proof of The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$ takes from Carathéodory's characterisation: $f$ is differentiable at $c$ if and only if there is $\phi :A\to \mathbb{R}$, continuous at $c$, with $f(x)-f(c)=\phi (x)(x-c)$ for every $x\in A$, and then $\phi$ is unique and $\phi (c)=f^{\mathrm{\prime }}(c)$ is, for a power, an explicit polynomial, and no auxiliary case distinction is hidden inside it.

• The closed form of the factor is what makes claim 2 worth stating. For a general $f$ the Carathéodory factor is produced by Carathéodory's characterisation: $f$ is differentiable at $c$ if and only if there is $\phi :A\to \mathbb{R}$, continuous at $c$, with $f(x)-f(c)=\phi (x)(x-c)$ for every $x\in A$, and then $\phi$ is unique and $\phi (c)=f^{\mathrm{\prime }}(c)$ out of the difference quotient itself, and its value at the base point is filled in by hand; for a power it is a polynomial written down in advance, by Factorisation of $b^n-a^n$, and the resulting Lipschitz estimate, and its continuity is then a reading of Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function rather than an appeal to the derivative. The two routes agree, which is step 2.1.

• Nested composites cost nothing extra. Claim 3 applies The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$ twice, and at each application the inner function maps $\mathbb{R}$ into $\mathbb{R}$, so the hypothesis that the image point is a limit point of the outer domain is automatic. On a smaller domain it would not be, and that is the hypothesis a careless nesting would drop.

• What the numerals hide. $\iota (18)=\iota (3)\iota (6)=\iota (3)\iota (2)\iota (3)$ is an identity of canonical naturals (Canonical naturals are positive and strictly increasing), not an arithmetic fact about the symbol $18$; every collapse of a product of numerals above is that lemma.