I've been recently interested in the nth derivative of $x^x$, and I jumped to pondering of the inverse of not this function, (because it's known to be $e^{\operatorname{W}\left(\ln x\right)}$, for $\operatorname{W}\left(x\right)$ being the Lambert W Function) but rather a more generalized one: $x^{x+k}$. I would look towards using the Lagrange Inversion Theorem, but this function is non-analytic. Maybe we could use $(x+k)^{(x+k)}$, but nonetheless, I'm stuck. Wolfram|Alpha gives way too much data in terms of solving $\left(ax+b\right)^{\left(cx+d\right)}=0$, and asking Wolfram|Alpha to invert the function doesn't exactly work, either. Thanks!
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1$\left(ax+b\right)^{\left(cx+d\right)}=0$ only if $ax+b = 0$. I did not say also "if" because of complications like $0^0 = 1$ and $0^{-1}$ is undefined, and so on. – GEdgar Jul 02 '20 at 23:11
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1This post inverts $(a+bx)^x$ – Тyma Gaidash Jun 15 '23 at 18:13