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differentiation rules for x[4]n, where n is any natural number

Looks like there is a recurrence relation here, but I don't know how to write it for d/dx x[4]n cuz all the stuff that is multiplied into the tetration is nested, so you have to express that with a function...

How does one express A_n(x) in closed form? I really don't know where i am going with this, perhaps doing A_k(x) at non-integer k to obtain the values for x[4]a for any positive real a?
EDIT oh, I have found it... , so . but i am suspicious about its utility for all real numbers, because it is not true for n=1:

there seems to be a great gap between x[4]1 and x[4]2.
heck yeah, there IS a great gap, because x[4]2 is transcendental, while x[4]1 is simply the polynomial x

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differentiation rules for x[4]n, where n is any natural number - by Base-Acid Tetration - 05/23/2009, 04:04 AM

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