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

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.
