General Power Rule: Proof & Real Values of x?

[reinloch]

Hi all, regarding the proof of the general power rule,

If we let $y = x^r$, then $\ln y = r\ln x$, and then by implicit differentiation
$\frac{y'}{y} = \frac{r}{x},$
and thus it follows that
$y' = \frac{ry}{x} = \frac{rx^r}{r} = rx^{r-1}.$

But the statement $\ln y = r\ln x$ also requires $x>0$, so does the General Power Rule only applies to positive real values of x?

Thanks.

 

[mfb]

For negative real values of x, x^r is well-defined for integer r only.
In those cases, you can modify the proof a bit to work with negative values, too.