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

by RoRi

November 15, 2015

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

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

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

Proof. We write
$f(x) = x^r = e^{r \log x}.$

Therefore,

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

August 22, 2017 at 7:56 am

Johannes says:

I added that the formula is only valid for rational r if x≤0, since this has already been proven by other means earlier in the book.

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