- #1
japplepie
- 93
- 0
d(x^^n)/dx = ?
What is the derivative of a sexp()
- Context: Undergrad
- Thread starter japplepie
- Start date
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- Derivative
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Discussion Overview
The discussion revolves around the derivative of the function represented by tetration, specifically the expression \(x^^n\), which denotes a power tower of height \(n\) with base \(x\). Participants explore the mathematical properties and challenges associated with differentiating this function.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants define \(x^^n\) as tetration, explaining it as a power tower of \(x\) raised to itself \(n\) times.
- There is uncertainty regarding the existence of a simple solution for the derivative of \(x^^n\).
- One participant suggests that the function may be simpler for integer values of \(x\), while another counters that it is not defined for integer values, later clarifying to positive integers.
- Another participant proposes a rewriting of the function using the exponential and logarithmic forms, indicating that differentiation would need to be performed manually.
- A follow-up question is posed regarding the convergence of the limit of the \(i\)-th derivative of \(x^^n\) as \(i\) approaches infinity.
Areas of Agreement / Disagreement
Participants express differing views on the definition and properties of \(x^^n\), particularly regarding its applicability to integer values. The discussion remains unresolved with multiple competing perspectives on the differentiation of the function.
Contextual Notes
There are limitations in the discussion regarding the assumptions about the domain of \(x\) and the mathematical steps involved in differentiating \(x^^n\). The nature of tetration and its derivatives appears to depend on specific definitions and conditions that are not fully explored.
- #2
moriheru
- 273
- 16
What is (x^^n)?
- #3
Mentallic
Homework Helper
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moriheru said:
It's tetration which represents a double up arrow in Knuth's up arrow notation. Basically a power tower of x's n high:
[tex]x^{x^{.^{.^{.^x}}}}[/tex]
And I don't see any simple solution to this problem.
- #4
japplepie
- 93
- 0
Im only looking for the formula for integer values of x, will that make it simpler?Mentallic said:
- #5
HallsofIvy
Science Advisor
Homework Helper
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That function is not even defined for integer values of x.
- #6
japplepie
- 93
- 0
Sorry, what I meant was positive integers.HallsofIvy said:
- #7
DarthMatter
- 94
- 10
You can rewrite ## x^x = \exp(\ln(x)\cdot x) ##. similar ##x^{x^x}=\exp(\exp(\ln(x)\cdot x)\cdot \ln(x)) ##. But you'll have to derivate yourself, I'm too lazy right now. 
- #8
japplepie
- 93
- 0
ok I got it
d(x ^^ n) / dx = x ^^ n * d(x ^^ ( n -1) * ln x ) / dx
Follow up question:
lim i→ ∞ { di(x ^^ n) / dxi } converge?
d(x ^^ n) / dx = x ^^ n * d(x ^^ ( n -1) * ln x ) / dx
Follow up question:
lim i→ ∞ { di(x ^^ n) / dxi } converge?
Last edited:
- #9
Raghav Gupta
- 1,010
- 76
It would be better if you type in latex.japplepie said:
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