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I'm confused about something regarding the Power Rule. The Power Rule states that:

$f(x) = x^n$, $n ≠ 0$
then
$f'(x) = nx^{n-1}$

So it doesn't say anything regarding the x. Then if I take x=0 and n=1, it should read as follows:

$f'(0) = 1\cdot0^{0}$ which is as far as I know undefined. Shouldn't that give me the correct answer 0? What am I missing here? Thank you very much in advance! Best Regards

charlietan84
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  • It also goes wrong at $x = 0$ when $n < 0$. You need to handle it as a special case. – badjohn Apr 10 '23 at 11:51
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    in the context of polynomials, you should interpret $x^0$ and $0^0$ as 1 – Matthew Towers Apr 10 '23 at 11:52
  • Don't forget that things should make sense given their "meaning," and this often guides convention. Clearly the slope of $f(x)=x$ at $x=0$ is $1$ (and so the same is true of its tangent line), so this should mean $1$. It shouldn't mean $0$ because the slope of $f(x)=x$ is $1$ everywhere, not $0$. – Randall Apr 10 '23 at 12:02
  • Going back to the basic definitions here is best. Things like "the power rule" or other shortcuts are merely there as useful memory tools and results for certain situations but are not themselves "definitions" or anything that must be strictly adhered to. If you are confused about what the derivative of a constant function is... go back to the $f'(x) = \lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$ definition to which point it should be obvious that the derivative of a constant function is everywhere zero. Similarly you can find the derivative of a linear function easily as well. – JMoravitz Apr 10 '23 at 12:07

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