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Show that the formula for rational powers holds for real powers of positive numbers

If r \in \mathbb{Q} and x > 0 we proved that

    \[ f(x) = x^r \quad \implies \quad f'(x) = rx^{r-1}. \]

  1. Prove the above formula for f'(x) is valid for any r \in \mathbb{R}.
  2. If x \leq 0 when does the above formula hold?

  1. Proof. We write

        \[ f(x) = x^r = e^{r \log x}. \]

    Therefore,

        \[ f'(x) = \left( e^{r \log x} \right) \left( \frac{r}{x} \right) = x^r \left(\frac{r}{x} \right) = rx^{r-1}. \qquad \blacksquare\]

  2. The problem with this formula if x \leq 0 is that \log x is not defined there. I’m not sure what else Apostol wants us to say other than we can not meaningfully write x^r = e^{r \log x} if x \leq 0.

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