How can i differentiate this ?

In summary, the conversation discusses the power tower function y(x)=x^{x^{x...}} and its derivative for different values of x. The upper limit for a power tower is approximately 1.44467, and the conversation also mentions using larger values such as x^2 and above. It is concluded that the function is 0 for x<1, 1 for x=1, and infinity for x>1, with a derivative of 0 for 0≤x≤1 and undefined for x>1. The possibility of graphing the function is also brought up as a potential challenge.
  • #1
abia ubong
70
0
I wonder if this can be differentiated ,if it can then what is the derivative of
[x^x^x^x^x^x...]^[(x^2)^(x^2)^(x^2)...]^[(x^3)^(x^3)^(x^3)...]^[(x^4)^(x^5)...]......
thank you
 
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  • #2
what do all the dots mean? What order do you want the exponentiation in (it's not associative)?
 
  • #3
From left to right i guess.I assume it resembles power tower.


Daniel.
 
  • #4
The power tower:

[tex]y(x)=x^{x^{x^. . .}}}[/tex]

is defined for:

[tex]e^{-e}\leq x\leq e^{1/e}[/tex]

Note that the upper limit for a power tower is about 1.44467. However above, you're using "2" and larger values, you know [itex]x^2[/itex] and more. Thus I claim the expression is 0 for x<1, 1 for x=1, and infinity for x>1. It's derivative therefore is zero for [itex]0\leq x\leq 1[/itex] and undefined for x>1.

However, I've just started looking at it and wouldn't bet more than a dollar.
 
  • #5
Here's a better question - "Can you draw the graph of it ?"

If you cannot draw a graph of that thing, then chances are pretty good that you can't differentiate it either.
 

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