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Firstly, yes, I know how to expand expressions like $(a+b)^{({1}/{2})}$ but I want a rigorous proof of why it is okay to replace binomial coefficients like ${1/2}\choose{2} $ with $({1/2})({1/2-1}))/2$

(Of course, I already understand the proof of binomial theorem for integers using combinatorics or induction)

A similar question was posed on this site earlier, but I don't feel the answers really get to the bottom of my doubt:

Binomial theorem for non integers ? O_o ??

For example, one of the answers cited tailor's theorem from calculus. My problem with that is that my understanding of derivatives of monic polynomials mainly comes from using the binomial theorem to expand $(x+\epsilon)^n$ for arbitrarily small $\epsilon$

Thus, this felt like a circular proof to me.

So I would either like:

A basic proof of the binomial theorem without derivatives

OR

An alternative understanding of derivatives of $x^n$ without binomial expansions

aman
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  • The rule $\dfrac{d}{dx}x^{\alpha} = \alpha x^{\alpha-1}$ works for all real numbers $\alpha$ (and whenever $x^{\alpha}$ is defined). Note that if the exponent $\alpha$ is not an integer, then one of the ways to define it is $x^{\alpha} := e^{\alpha \ln(x)}$ (so we require $x > 0$). So, applying Taylor's theorem is definitely not circular, once you have properly established a definition for the symbol $x^{\alpha}$ for positive $x$, and any real number $\alpha$. – peek-a-boo May 02 '20 at 06:05
  • I don't understand how the $nx^{n-1}$ works for non-integers. The proof for this expression relies on binomial theorem right? – aman May 02 '20 at 06:08
  • Yes, I guess I don't really understand what raising a number to any real power means. Can you help me with this? – aman May 02 '20 at 06:10
  • That's the proof of the baby version, which is usually presented first in elementary classes. But it is true in general. The first question you should be asking isn't "why is $\dfrac{d}{dx}(x^n) = nx^{n-1}$". It should be "what's the meaning of $x^n$ if $n$ is not an integer or rational number". And for this, refer to my previous comment. One has to first define the exponential function $\exp: \Bbb{R} \to [0, \infty)$ (either using a power series definition, or for example, by first defining the logarithm, $\ln(x):= \int_1^x \dfrac{1}{t} , dt$, then defining $\exp$ to be the inverse of $\ln$) – peek-a-boo May 02 '20 at 06:12
  • in any case, once you define $\exp$ and $\ln$, you can then define the symbol $x^{\alpha}$, for any positive number $x$, and any real number $\alpha$. Then, you can apply the chain rule to prove that differentiation rule. Anyway, the questions you're asking are very good, but they're not easy to answer if you don't have the proper calculus background to understand the various definitions. But if you do, then you should take a look at any good calculus text, for example, Michael Spivak's Calculus. – peek-a-boo May 02 '20 at 06:16
  • you define $x^{\alpha} := e^{\alpha \ln(x)}$ (i.e the RHS is a known quantity, and you're using it to define the LHS). And in the case of integer $\alpha$, this will coincide with the usual definition – peek-a-boo May 02 '20 at 06:21
  • Here is my understanding so far: You first define $ln$ as the function with an instantaneous rate of change as $1/x$ Then, you define $e^x$ to be the inverse of that function. Now, you know nothing whatsoever about the properties of this function (why is it its own derivative, etc) So how do you move on to differentiating $x^n$ – aman May 02 '20 at 06:23
  • In fact, how do you even know that $x^n=e^{nln(x)}$ from this definition... – aman May 02 '20 at 06:25
  • you have to establish all of those results along the way, in a systematic fashion of course. I'm not claiming these results are obvious by any means. There is an entire chapter of a textbook devoted to this, so there's far too much information for me to summarize in a short stackexchange discussion. I really suggest you to take a look at the textbook I referenced. All your questions will be answered there. – peek-a-boo May 02 '20 at 06:27
  • Ok I don't know if I'm ready for it though. Wouldn't it be easier to define $x^k$ as the limit of the products of the exponents in decimal notation like $x^\pi=x^3*x^{0.1}...$ – aman May 02 '20 at 06:59
  • Then one can somehow figure out derivative of $x^{1/n}$ and using chain rule $x^{a/b}$ From this we can work for some kind of limit? Maybe I am being too simplistic? – aman May 02 '20 at 07:02
  • There's no easy way to show this for fractional exponents without calculus. – Allawonder May 02 '20 at 07:10
  • You can read various approaches to general binomial theorem here and here. A typical proof (first link) uses the derivative formula for $x^n$ which holds for all real $n$. Second link is based on more intuitive idea. – Paramanand Singh May 02 '20 at 16:05
  • For derivative of $x^n$ with rational $n$ see https://math.stackexchange.com/a/1782225/72031 – Paramanand Singh May 02 '20 at 16:10
  • Your approach to definition of $x^n$ when $n$ is irrational is handled in this post. – Paramanand Singh May 02 '20 at 16:14

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