Exploring Pentation - Base e

[quickfur]

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I would like to "integrate" (or differentiate) e.g pentation to obtain "slower" operation (or faster).
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Given any function f(x) that grows asymptotically faster than \( e^x \), the derivative f'(x) must necessarily grow faster than f itself. The converse is also true: if f(x) grows asymptotically slower than \( e^x \), then f'(x) will grow asymptotically slower than f itself. When this difference is precisely by a factor of x (i.e., \( \lim_{x\rightarrow\infty}f(x)/f'(x)=Cx+D \) for constants C≠0, D), then f(x) is a polynomial. There exist functions whose asymptotic ratio with their derivatives lie between 0 and Cx+D... I'll leave this as an exercise for the reader.

Since tetration, pentation, and higher grow a lot faster than exponentiation, their derivative must necessarily be faster. Moreover, the difference between the derivative of a pentation and pentation will be much greater than the difference between the derivative of tetration and tetration.