I General Nth Derivative f(x)=x^x: Solving a Difficult Problem

[Euler2718]

I'm very interesting in functions of the nature:

f(x) = x^{x}
f(x) = x^{x^{x}}
and so on. I believe these are called tetrations? Regardless, I sought to generalize the nth derivative of f(x)=x^x and it is proving to be difficult.

First I tried just repeatedly differentiating until I could see a pattern:

f'(x) = x^{x}\left(1+\ln(x)\right)
f''(x) = x^{x-1}\left(1+x+2x\ln(x) + x\ln^{x}(x)\right)
f'''(x) = x^{x-2}\left( -1+3x+x^{2} +3x(1+x)\ln(x)+3x^{2}\ln^{2}(x)+x^{2}\ln^{3}(x)\right)
f^{4}(x) = x^{x-3}\left( 2-x+6x^{2}+x^{3}+4x(-1+3x+x^{2})\ln(x)+6x^{2}(1+x)\ln^{2}(x)+4x^{3}\ln^{3}(x)+x^{3}\ln^{4}(x) \right)

I couldn't seem to find a pattern here. So I tried representing f(x)=x^{x} as a power series:

f(x) = x^{x} = e^{x\ln(x)} = \sum_{n=0}^{\infty} \frac{1}{n!}x^n\ln^{n}(x)

Then maybe taking the derivatives of the power series could lead me to an easier pattern. After I took the first derivative I felt that it wasn't the case as:

f'(x) = \sum_{n=0}^{\infty} \frac{1}{(n-1)!}\left(1+\ln(x)\right)x^{n-1}\ln^{n-1}(x)

Which looks to be at a first glance more complex. Putting it through mathematica kind of left me with the same feeling of getting no where.

Is there anything I should be considering? Any methods that I could employ to solve my problem? I don't know much about higher level maths (I'm currently doing Calc III) .

 

[Simon Bridge]

what is wrong with:
$y=x^x$
$\ln |y| = x\ln |x|$
$\qquad$... differentiate both side wrt x, and solve for y'

http://www.analyzemath.com/calculus/Differentiation/first_derivative.html

 

[Euler2718]

what is wrong with:
$y=x^x$
$\ln |y| = x\ln |x|$
$\qquad$... and differentiating both side wrt x.

http://www.analyzemath.com/calculus/Differentiation/first_derivative.html

I'm interesting personally in a formula that would give me any derivative. Say I wanted the 10th derivative for instance. Of course you could manually take d/dx 10 times, but I think a formula would be a nice discovery.

 

[Simon Bridge]

Oh I getcha...
I'd try plotting the successive analytic derivatives to see if the curves show any patterns.
The alternative would be to prove there wasn't one.

You are getting $f^{(n)} = x^{x-n+1}\left(\sum_{i=0}^n P_i^{n-1}(x) \ln^i|x|\right)$ ... something... where $P_i^j(k)$ is the ith polynomial in k of degree j.
... except the pattern breaks for n=4 with P^3,P^6,P^3,P^6,P^3 ... but this may point to a way to find a pattern.

You may want to get a computer to generate a lot of derivatives.
Good luck.

 

[Euler2718]

I've gotten to a point to where if I could generalize the nth derivative of

f(x) = \ln^{n}(x)

I might be on a good path. This generalization looks messy though.

 

[Ssnow]

Using the general product rule:

$D^{m} f(x)=D^{m}\left(\sum_{n=0}^{\infty}\frac{1}{n!}x^{n}\ln^{n}{x}\right)=\sum_{n=0}^{\infty}\frac{1}{n!}D^{m}\left(x^{n}\ln^{n}{x}\right)=$

$=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{i=0}^{m}\binom{m}{i}D^{i}(x^{n})D^{m-i}(\ln^{n}{x})$

put it into a math program and it generates the $m$-derivatives ...