General Power Rule

  • Thread starter reinloch
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  • #1
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Hi all, regarding the proof of the general power rule,

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

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

Thanks.
 

Answers and Replies

  • #2
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For negative real values of x, xr is well-defined for integer r only.
In those cases, you can modify the proof a bit to work with negative values, too.
 

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