Editing Power rule (section)

==Generalizations==

===Complex power functions===

If we consider functions of the form <math>f(z) = z^c</math> where <math>c</math> is any [[complex number]] and <math>z</math> is a complex number in a slit complex plane that excludes the [[branch point]] of 0 and any branch cut connected to it, and we use the conventional multivalued definition <math>z^c := \exp(c\ln z)</math>, then it is straightforward to show that, on each branch of the complex logarithm, the same argument used above yields a similar result: <math>f'(z) = \frac{c}{z}\exp(c\ln z)</math>.<ref>{{cite book|last1=Freitag|first1=Eberhard|last2=Busam|first2=Rolf|title=Complex Analysis|date=2009|publisher=Springer-Verlag|location=Heidelberg|isbn=978-3-540-93982-5|page=46|edition=2}}</ref>

In addition, if <math>c</math> is a positive integer, then there is no need for a branch cut: one may define <math>f(0) = 0</math>, or define positive integral complex powers through complex multiplication, and show that <math>f'(z) = cz^{c-1}</math> for all complex <math>z</math>, from the definition of the derivative and the binomial theorem.

However, due to the multivalued nature of complex power functions for non-integer exponents, one must be careful to specify the branch of the complex logarithm being used. In addition, no matter which branch is used, if <math>c</math> is not a positive integer, then the function is not differentiable at 0.