- #1
japplepie
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d(x^^n)/dx = ?
What is the derivative of a sexp()
- Thread starter japplepie
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In summary: Anyway, the answer is no, it does not converge. However, for fixed n it does converge.In summary, the conversation discusses tetration, which represents a power tower of x's n high in Knuth's up arrow notation. The formula for integer values of x is discussed, but it is not defined for integer values of x. The conversation also mentions a possible solution using logarithms, but it is not provided. In response to a follow-up question, it is stated that the limit does not converge, but for fixed n it does converge.
- #2
moriheru
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What is (x^^n)?
- #3
Mentallic
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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
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Im only looking for the formula for integer values of x, will that make it simpler?Mentallic said:
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HallsofIvy
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That function is not even defined for integer values of x.
- #6
japplepie
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Sorry, what I meant was positive integers.HallsofIvy said:
- #7
DarthMatter
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You can rewrite ## x^x = \exp(\ln(x)\cdot x) ##. Similiar ##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
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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
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It would be better if you type in latex.japplepie said:
Related to What is the derivative of a sexp()
What is the derivative of a sexp()?
The derivative of a sexp() is a mathematical concept used in calculus to represent the rate of change of a function with respect to its independent variable.
How is the derivative of a sexp() calculated?
The derivative of a sexp() can be calculated using the rules of differentiation, which involve taking the limit of the change in the function over the change in the independent variable as the change approaches 0.
What is the significance of the derivative of a sexp()?
The derivative of a sexp() has many practical applications in science and engineering, as it allows us to analyze the behavior of functions and make predictions about their future values.
Can the derivative of a sexp() be negative?
Yes, the derivative of a sexp() can be negative, as it represents the slope of the function at a specific point. A negative value indicates a decreasing function, while a positive value indicates an increasing function.
Is the derivative of a sexp() the same as the slope of the tangent line?
Yes, the derivative of a sexp() is equal to the slope of the tangent line at a specific point on the function. This is because the tangent line represents the instantaneous rate of change of the function at that point, which is what the derivative measures.
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