Nth derivative of $n^n$ I believe the limit derivative of $n^n$ is $n!$ where the limit derivative of a function is the derivative that will turn that function into a constant.
For polynomial functions, the limit derivative is the $m$th derivative where $m$ is the order of the function.
I discovered that for functioms of the form $n^c$ the limit derivative was $c!$ where $c$ was a known positive integer constant. Can someone prove or disprove this.
 A: Since the derivative of $x^x$ is $x^x+x^x\ln x$,


*

*$\frac{d}{dx}x^x=x^x+x^x\ln x$.  Plugging in $x=1$ gives $1^1+1^1\ln(1)=1=1!$, so your formula works for the first derivative.

*$\frac{d^2}{dx^2}x^2=\frac{d}{dx}(x^x+x^x\ln x)=x^x+2x^x\ln x+x^x\ln^2x+x^{x-1}$.  Plugging in $x=2$ gives $2^2+2(2)^2\ln 2+2^2\ln^2(2)+2^1=6+8\ln 2+4\ln^2(2)$, which is not $2!$.  Therefore, your formula fails for $n=2$.
It is true, however, that


*

*$\frac{d^m}{dx^m}x^m=m!$ for all $m>0$.  The easiest proof is via induction where $\frac{d^m}{dx^m}x^m=m\frac{d^{m-1}}{dx^{m-1}}x^{m-1}=m(m-1)!=m!$.

A: You are looking for "the derivative that will turn that function into a constant".
Let 
$$f^{(k)}=C.$$ Then by successive integrations,
$$f^{(k-1)}=Cx+C',$$
$$f^{(k-2)}=\frac12Cx^2+C'x+C'',$$
$$\cdots$$
Continuing this way, all you get are polynomials, which are the only functions with an eventually constant derivative, so your question has no answer.
By the way, finding a closed formula for the derivatives of $x^x$ looks somewhat hopeless.

A: if you are asking for the derivative of the map, $f: \mathbb{R}_{+}^* \to \mathbb{R}$, defined by $f(x) = x^x = \exp(x\ln(x))$. You get : 
\begin{equation*}
 f'(x)= x^x (\ln(x) + 1) \quad f''(x)= x^x \left(\frac{1}{x} + \ln(x) + 1\right) \; \cdots
\end{equation*}
The $n$-$th$ derivative of $f$ is never constant, because $f$ is not a polynomial function (it grows much faster than any polynomial)
Otherwise if you are considering only sequences, you can define the discrete derivative as follows. If $(u_n)$ is a sequence of real numbers, the discrete derivative of $u$, denoted by $\delta u$, is defined by $(\delta u)_n = u_{n+1} - u_n$. But even when using this definition, the limit derivative of $n^n$ is not defined properly, because the discrete derivatives of the sequence $(n^n)$ are never constants sequences... 
