 differentiation rules for xn, where n is any natural number - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: differentiation rules for xn, where n is any natural number (/showthread.php?tid=294) differentiation rules for xn, where n is any natural number - Base-Acid Tetration - 05/23/2009 Looks like there is a recurrence relation here, but I don't know how to write it for d/dx xn 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 xa 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 x1 and x2. heck yeah, there IS a great gap, because x2 is transcendental, while x1 is simply the polynomial x RE: differentiation rules for xn, where n is any natural number - bo198214 - 05/23/2009 (05/23/2009, 04:04 AM)Tetratophile Wrote: 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 x1 and x2. heck yeah, there IS a great gap, because x2 is transcendental, while x1 is simply the polynomial x Your recursion formula is wrong, correctly it should be: or equivalently: then or with your notation RE: differentiation rules for xn, where n is any natural number - Base-Acid Tetration - 05/25/2009 the problem is now extending our function A_n to non-natural n... RE: differentiation rules for xn, where n is any natural number - bo198214 - 05/25/2009 (05/25/2009, 09:31 AM)Ansus Wrote: Great! I've added the rule to our wiki page http://en.wikipedia.org/wiki/User:MathFacts/Tetration_Summary This works also for arbitrary complex : RE: differentiation rules for xn, where n is any natural number - andydude - 05/26/2009 This derivative is also what I used in this thread, which no one seemed to notice. Yes, Ansus' formula is correct, although usually sexp=tet, so I would not use that notation. I would use the notation Andrew Robbins