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power function
A real power function $f\!:\,\mathbb{R}_+\to\mathbb{R}$ has the form $$f(x) \;=\; x^a$$ where $a$ is a given real number.
Theorem 1 The power function $x\mapsto x^a$ is differentiable with the derivative $x\mapsto ax^{a-1}$ and strictly increasing if $a > 0$ and strictly decreasing if $a < 0$ (and constant 1 if $a = 0$ ).
The power functions comprise the natural power functions $x\mapsto x^n$ with $n = 0,\,1,\,2,\,\ldots$ , the root functions $x\mapsto \sqrt[n]{x} = x^{\frac{1}{n}}$ with $n = 1,\,2,\,3,\,\ldots$ and other fraction power functions $x\mapsto x^a$ with $a$ any fractional number.
Note. The power $x^a$ may of course be meaningful also for other than positive values of $x$ , if $a$ is an integer. On the other hand, e.g. $(-1)^{\sqrt{2}}$ has no real values -- see the general power.
power function is owned by J. Pahikkala.
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