[nvzmrton] Derivative of super logarithm or iterated logarithm

What is the derivative of the superlogarithm or iterated logarithm lg*(x), the inverse of tetration?

Motivation is the derivative of log(x) is 1/x, yielding the harmonic series which slowly diverges as shown by the integral test.  The derivative of log(log x) is 1/(x log x) yielding a very slowly diverging series, and so forth.

Also of interest is the derivative of (lg* x)^-1, which would yield a very slowly converging series.

For lg*, we could define something piecewise, perhaps a step function, but the hope is something more elegant and meaningful.

And then, the derivative of the inverse Ackermann function?