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Differentiating the functions $x^x$, $x^{x^x}$ (or ${^2{x}}$ and ${^3{x}}$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this forum) a general expression for the derivative of ${^n{x}}$. However, I would like to find the derivative ${^x{x}}$, or $x$ tetrated to the $x$. I seem to have absolutely no tools to even approach this problem, as I am not at all familiar with the maths associated with tetration.

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    $\begingroup$ How do you define ${^y{x}}$ for $y \in\Bbb R$? $\endgroup$ – Martin R Nov 13 at 8:08
  • $\begingroup$ Good point. It seems that there is no extension into the reals for tetration. $\endgroup$ – A. Lavie Nov 14 at 17:15
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$$\frac{d}{dx}(^nx) = (^nx)\Bigl((lnx)\frac{d}{dx}(^{n-1}x)+\frac{^{n-1}x}{x}\Bigl)$$

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    $\begingroup$ I am talking about the derivative for ${^x{x}}$, not $x^x$. $\endgroup$ – A. Lavie Nov 6 at 3:40

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