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**simplependulum** For a postive integers $\displaystyle n$

we have $\displaystyle ( x^n)' = nx^{n-1} $

for a fraction $\displaystyle \frac{p}{q} $

we also have $\displaystyle (x^{\frac{p}{q}})' = \frac{p}{q} x^{\frac{p}{q} -1} $

I can complete the above proofs but don't know to prove for irrational numbers index $\displaystyle n $ ? Is $\displaystyle (x^{n})' = n x^{n-1} $ ?

Moreover , are $\displaystyle (x^{\pi} )' = \pi x^{\pi -1} $ and $\displaystyle (x^{e} )' = e x^{e -1} $ ?

Thank you !