How do I find the derivative of pentation?

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SUMMARY

The discussion focuses on finding the derivative of the first pentation function, specifically x pentated to x. Pentation, recognized as the 5th hyperoperation, follows tetration, which is defined as x tetrated n times (x^x^... n times). Unlike tetration, which can be expressed in terms of exponentiation and is manageable in tools like Wolfram Alpha, pentation cannot be easily notated or derived using standard graphing programs. The suggestion to take the natural logarithm of both sides multiple times is proposed as a potential method for approximation.

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Ramanujan
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How do I find the derivative of the first pentation function, i.e. x pentated to x. i can't even notate this in my graphing program so i don't know how to ask it to derive this function. pentation is the 5th hyperoperation, coming after tetration which is after exponentiation etc. x tetrated to n is x^x^x n times. tetration can be written as x^x and thus grapher and wolfram alpha can derive this function. pentation on the other hand, is x tetrated n times, which can not be written as a function in terms of multiplication or exponentiation, and thus i have no way of telling graphing programs what it is as a function.
 
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How about if you take the natural log of both sides approximately n times?
 

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