关于根式函数与分数指数幂函数求导的严谨性问询

Hey folks, I’m a math student with only a basic background in real analysis (haven’t touched complex analysis at all yet), and I’ve got a question about derivatives of radical functions—specifically related to the fractional exponents we learned in high school, but I’m hoping for a more rigorous, formal explanation than what we got back then.

Let me start with a quick recap of the high school stuff to set the stage: we were told that for positive bases, fractional exponents can be rewritten as radicals, and all the exponent rules hold as expected. But non-positive bases are a whole different story, and here’s a classic counterexample that shows why mindlessly applying those rules leads to trouble:

$$ (-1)^{{\frac{1}{3}}=(-1)}{\frac{2}{6}}=((-1)²⁾{\frac{1}{6}}=1^{\frac{1}{6}}=1 $$

The first two equalities use the exponent rules we learned, but if we accept them, we end up with a contradiction—since we know the cube root of -1 is -1, not 1. Clearly, we can’t just freely rewrite $(-1)^{\frac{1}{3}}$ as $\sqrt[3]{-1}$ and apply exponent rules when the base is negative; those rules have strict limitations outside positive bases.

Now I’m moving into calculus, and I’m curious about the power rule for differentiation when it comes to these fractional exponent/radical functions. We learn the formula that if $f(x) = x^r$, then $f'(x) = r x^{r-1}$—but I want to understand the rigorous conditions for this when $r$ is a fraction, especially since we saw how messy non-positive bases get with exponents.

Specifically, I’m wondering:

• What’s the precise domain in real analysis where the power rule is valid for fractional exponents?
• How do we properly link the derivative of a radical function (like $\sqrt[3]{x}$ or $\sqrt{x}$) to its fractional exponent form without running into contradictions similar to the negative base example above?
• Are there any subtle edge cases or restrictions I need to keep in mind when differentiating these functions that go beyond just “stick to positive bases”?

I’d really appreciate any insights or formal explanations that align with basic real analysis—no complex analysis jargon, please!

备注：内容来源于stack exchange，提问作者Timo Chang