Does the power rule come from the generalised binomial theorem or the other way around?

Question

Remember how we say $lim \frac{\sin{x}}{x}=1$ doesn't come from the L Hospital's rule, because the differentiation of $sin x$ from first principles uses the fact that $lim \frac{\sin{x}}{x}=1$?

A similar situation is arising for the power rule and the generalised binomial theorem.

The generalised binomial theorem can be proven as the Taylor expansion of $(1+x)^r$. This derivation uses the power rule, as we need the power rule to calculate derivatives of $(1+x)^r$

The derivation of the power rule from first principles would need the generalised binomial theorem, as we'd need to expand the term $(x+h)^r$

So which of these two results is more fundamental? Which one comes from the other and why?

Answer 1

I'm guessing:

$\lim_{h\rightarrow 0} \frac{(x+h)^r-x^r}{h}$

=$\lim_{h\rightarrow 0} \frac{e^{r\ln{x+h}}-x^r}{h}$

Now we use the L Hospital's rule

$=\lim_{h\rightarrow 0} e^{r\ln{x+h}} r \frac{1}{x+h}$

$= rx^{r-1}$

So the generalised power rule does not depend on the generalised binomial theorem.

Answer 2

The power rule comes from the generalised binomial theorem.

Generalised binomial theorem: ($1665$)

We find in section 1.2 Exponentials and the Binomial Theorem of Analysis by Its History by E. Hairer and G. Wanner a citation from Newton

• [Newton:] All this was in the two plague years of $1665$ and $1666$, for in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any other time since.

  followed by

  One of Newton's ideas of these anni mirabiles, inspired by the work of Wallis was to try to interpolate the polynomials $(1+x)^0, (1+x)^1, (1+x)^2,\ldots$ in order to obtain a series for $(1+x)^{\alpha}$ where $\alpha$ is some rational number. This means that we must interpolate the coefficients given in \begin{align*} (1+x)^{n}=1+\frac{n}{1}x+\frac{n(n-1)}{1\cdot 2}x^2 +\frac{n(n-1)(n-2)}{1\cdot2\cdot3}x^3+\cdots\qquad n=0,1,2,3,\ldots \end{align*}

  Since the latter are polynomials in $n$, it is clear that the result is given by the same expression with $n$ replaced by $a$. We therefore arrive at the general theorem.

  Theorem: (Generalized binomial theorem of Newton). For any rational $a$ we have for $|x|<1$ \begin{align*} \color{blue}{(1+x)^{a}=1+\frac{a}{1}x+\frac{a(a-1)}{1\cdot 2}x^2 +\frac{a(a-1)(a-2)}{1\cdot2\cdot3}x^3+\cdots} \end{align*}

From todays point of view this argumentation is (of course) not rigorous enough. The authors continue:

• Even Newton found that his interpolation argument was dangerous. Euler, in his Introductio ($1748$), stated the general theorem without any further proof or comment. Only Abel, a century later, felt the need for a rigorous proof.

Power rule: ($1670/1671$)

• Newton derived the power rule in his Method of Fluxions which was written in $1670/1671$ and published in $1736$.

  In a later treatise (german title: Abhandlung über die Quadratur der Kurven) from $1704$ he gave a more mature presentation of the Method of Fluxions where he also explicitly refers to the method of infinite series, i.e. to the generalised binomial theorem to derive the power rule.

A more detailed treatment of this treatise can be found e.g. in Grundlagen der Mathematik by Oskar Becker.

Side note: Limits (ca. $1850$)

• The usage of a limit as we use it today was not a theme at that time. It needed the rigor of Weierstrass and his school around $1850$.

  D. Hilbert wrote in Über das Unendliche ($1926$):

  It is essentially a merit of the scientific activity of Weierstrass that there exists at present in analysis full agreement and certainty concerning the course of such types of reasoning which are based on the concept of irrational number and of limit in general.

  We owe it to him that there is unanimity on all results in the most complicated questions concerning the theory of differential and integral equations, despite the most daring and diversified combinations with application of super-, juxta-, and transposition of limits.