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