Exploring the Graph and Applications of a^(x^x)

In summary, There is a function y=e^{x^{x^x}} that shows a rapid growth, much faster than an exponential function. Although it may not have practical applications, it has sparked interest in the field of BS-mathematics.
  • #1
2^Oscar
45
0
Hey guys,

I was wondering about what a graph would look like where the power to a did not increase at a linear rate i.e

a^(x^x)

Is there such a recognised function as this? If so does it have any practical applications and what does the graph look like?

Thanks,

Oscar
 
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  • #2


Hey 2Oscar! :smile:
2^Oscar said:
I was wondering about what a graph would look like where the power to a did not increase at a linear rate i.e

a^(x^x)

Is there such a recognised function as this? If so does it have any practical applications and what does the graph look like?

Never seen anything like it!

I'd be very surprised if it does have any practical applications.

Its graph would be like ax, only very much steeper. :smile:

Why don't you try working out its derivative? :wink:
 
  • #3


I'm unsure on how to differentiate a^x tbh... but i think i can do it for e^x...

so y= e^(x^x)

dy/dx = x2e^(x^x)?


My graphical calculator goes weird when i try to draw it lol... but i can see why it would be like a normal exponential graph just steeper.

Thanks,

Oscar
 
  • #4
2^Oscar said:
so y= e^(x^x)

dy/dx = x2e^(x^x)?

No …

e^(xx) = e^(exlogx),

so it's not x2, but d/dx(exlogx), = … ? :smile:

(and a^(xx) = e^(xxloga) wink:)
 
  • #5


sorry if I am misunderstanding... is the log to the base e?

If so then d/dx(e^xlogx) = (xlogx)e^(xlogx)?

Or perhaps d/dx(e^xlogx) = e^x2?

Sorry if i sound daft... not too familiar with this stuff :S


Thanks,

Oscar
 
  • #6
No, use the chain rule …

d/dx(exlogx) = exlogx d/dx(xlogx) :smile:

(use the X2 tag just above the reply box :wink:)
 
  • #7


so xlogx differentiates to 1+logx?

so (1+logx)exlogx?
 
  • #8
2^Oscar said:
so xlogx differentiates to 1+logx?

so (1+logx)exlogx?

Yup! :biggrin:
 
  • #9


Oh wow so the end differentiation is (1+logx)exlogxex^x?


Thanks so much for your help :D

Oscar
 
  • #10
You can simplify it a bit more …

(1+logx)xxex^x :wink:
 
  • #11


2^Oscar said:
Hey guys,

I was wondering about what a graph would look like where the power to a did not increase at a linear rate i.e

a^(x^x)

Usually when you look into powers that increase at a rate rather than linear, you try quadratic, not exponential :tongue2:

While I don't think it applies to anything, I'm more curious to dedicate my life studying:

[tex]y=e^{x^{x^x}}[/tex]

There is growing interest in the field of BS-mathematics :wink:

If you haven't noticed yet, it grows pretty fast.

[tex]x=1 ~ y=e[/tex]
[tex]x=1.5 ~ y=8[/tex]
[tex]x=2 ~ y=9,000,000[/tex]
[tex]x=2.3 ~ y>googol[/tex]
 

1. What is the significance of exploring a^(x^x)?

The function a^(x^x) is a type of exponential function known as a "tower function" or "tetration". It is a powerful mathematical tool that can represent extremely rapid growth and is useful in understanding complex systems and processes.

2. How is a^(x^x) graphed?

To graph a^(x^x), we first choose a base value for "a", which can be any positive number. Then, we plot points by choosing values for "x" and calculating a^(x^x). The resulting graph will have a steeply increasing curve that gets steeper and steeper as x increases.

3. What are some real-world applications of a^(x^x)?

A^(x^x) can be used to model population growth, spread of diseases, and the behavior of certain physical systems. It is also used in cryptography and computer science to represent extremely large numbers and for solving complex algorithms.

4. Can the graph of a^(x^x) ever decrease?

No, the graph of a^(x^x) can never decrease because the value of a^(x^x) will always be positive, regardless of the value of x. This is due to the nature of exponential functions, where the base is raised to a power, resulting in a positive output.

5. How does the value of "a" affect the graph of a^(x^x)?

The value of "a" affects the steepness and behavior of the graph. If a is less than 1, the graph will approach 0 as x increases. If a is greater than 1, the graph will increase rapidly. If a is equal to 1, the graph will be a straight line with a slope of 1. Additionally, a larger value of a will result in a steeper curve and a smaller value of a will result in a flatter curve.

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