What does the graph of a^(x^x) look like and what are its applications?

[2^Oscar]

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

 

[tiny-tim]

Hey 2^Oscar!

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 a^x, only very much steeper.

Why don't you try working out its derivative? ​

 

[2^Oscar]

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

 

[tiny-tim]

so y= e^(x^x)

dy/dx = x²e^(x^x)?

No …

e^(x^x) = e^(e^xlogx),

so it's not x², but d/dx(e^xlogx), = … ?

(and a^(x^x) = e^(x^xloga) wink:)

 

[2^Oscar]

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

 

[tiny-tim]

No, use the chain rule …

d/dx(e^xlogx) = e^xlogx d/dx(xlogx)

(use the X² tag just above the reply box )

 

[2^Oscar]

so xlogx differentiates to 1+logx?

so (1+logx)e^xlogx?

 

[tiny-tim]

so xlogx differentiates to 1+logx?

so (1+logx)e^xlogx?

Yup!

 

[2^Oscar]

Oh wow so the end differentiation is (1+logx)e^xlogxe^{x^x}?


Thanks so much for your help :D

Oscar

 

[tiny-tim]

You can simplify it a bit more …

(1+logx)x^xe^{x^x}

 

[Mentallic]

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

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

y=e^{x^{x^x}}

There is growing interest in the field of BS-mathematics

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

x=1 ~ y=e
x=1.5 ~ y=8
x=2 ~ y=9,000,000
x=2.3 ~ y>googol