Name and derivative of function $x^x$

Question

Name and derivative of function $x^x$

Power functions have base variable, but exponent fixed.

Exponential functions have base fixed, but exponent variable.

I can't find the function in title on derivatives tables.

Answer 1

According to WolframMathWorld functions of the type

$$a\uparrow\uparrow k\equiv \underbrace{a^{a^{\cdot^{\cdot^{\cdot^{a}}}}}}_{k}$$

are called power towers of $k$-th order also known as $k$-th tetration.

Therefore the function $f(x)=x^x$ could be called "power tower of order $2$" $($look here$)$.

Since it is not a that common function it would suprise me to find the derivative of this function in a standard table of differentiation. However, it is not hard to find the derivative of $f(x)$ by hand. Just consider that

$$x^x=e^{x\ln(x)}$$

and now by applying the rule for the derivative of an exponential combined with the chain rule we get

$$\begin{align} \frac d{dx} x^x&=\frac d{dx} e^{x\ln(x)}\\ &=e^{x\ln(x)}\cdot(x\ln(x))'\\ &=e^{x\ln(x)}\cdot(\ln(x)+1)\\ \end{align}$$

$$\therefore~ \frac d{dx} x^x=(\ln(x)+1)x^x$$

---

Furhtermore note that $x^x$ does not posses an elementery anti-derivative but the definite integral from $0$ to $1$ can be evaluated in a closed-form known as "Sophomore's Dream".

Answer 2

I don't know the name for $x^x$, but dervitate this function $f'(x)=f(x)(\ln f(x))'=x^x(x\ln x)'=x^x(\ln x+1)$