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The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the order of the derivative increases. I would like to find a closed formula that can calculate the root of the $n^{th}$ derivative, however I am having a difficult time finding any noticeable patterns. The best I have found is

$ a_N=\sum_{n=0}^{N-1} \left(\frac{(e-1)^n}{e\cdot n!} \right) $

where $N$ is the order of the derivative. This produces the first two roots, which are $\frac{1}{e}$ and $1$, but fails for any $N>2$. I assume that the zeros of the higher order derivatives have some relationship to $e$ but don't know what that relationship is. Any insights would be greatly appreciated.

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  • $\begingroup$ Here are the roots of the 2nd derivative, per Maple: 1, exp(2*LambertW(-(1/2)*exp(1/2))-1) $\endgroup$
    – Mark L. Stone
    Commented Jun 7, 2018 at 0:43
  • $\begingroup$ That is an interesting result, I was thinking only about the positive real roots but looking at the complex roots might give me a better understanding of what is going on between the derivatives. What about the third derivative? $\endgroup$
    – Nathaniel Weidman
    Commented Jun 7, 2018 at 0:54
  • $\begingroup$ Here are the numerical roots for 1st 30 derivatives per Maple fsolve. Rather multi-nodal. 0.3678794412 1. 1.594172705 2.156081546 2.693037878 0.7698317570 0.8922004122 1.076132692 1.297019230 1.551075081 1.836655380 2.149997442 2.484434915 2.832475855 3.188064676 3.547210496 3.907546791 4.267731451 6.550697400 4.984984445 5.341414895 3.785444605 3.679466558 3.837551474 4.023299304 4.227416876 4.446526524 4.679075468 4.924146178 5.180922002 $\endgroup$
    – Mark L. Stone
    Commented Jun 7, 2018 at 1:44
  • $\begingroup$ Very true, but is there anyway that an exact answer can be formulated. I know that some derivatives have more than one real root so in my studies I have focused on the larges root for each derivative. It may be that such a formulation doesn't exist; in which a proof would be accepted at an answer showing that it is impossible to find exact solutions. $\endgroup$
    – Nathaniel Weidman
    Commented Jun 7, 2018 at 2:32

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