# A^(x^x) graph

1. Jan 18, 2009

### 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

2. Jan 18, 2009

### tiny-tim

Re: a^(x^x)???

Hey 2Oscar!
Never seen anything like it! :surprised

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

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

Why don't you try working out its derivative?

3. Jan 18, 2009

### 2^Oscar

Re: a^(x^x)???

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 wierd 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. Jan 18, 2009

### tiny-tim

No …

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

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

(and a^(xx) = e^(xxloga) wink:)

5. Jan 18, 2009

### 2^Oscar

Re: a^(x^x)???

sorry if im 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. Jan 18, 2009

### tiny-tim

No, use the chain rule …

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

(use the X2 tag just above the reply box )

7. Jan 18, 2009

### 2^Oscar

Re: a^(x^x)???

so xlogx differentiates to 1+logx?

so (1+logx)exlogx?

8. Jan 18, 2009

### tiny-tim

Yup!

9. Jan 18, 2009

### 2^Oscar

Re: a^(x^x)???

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

Thanks so much for your help :D

Oscar

10. Jan 18, 2009

### tiny-tim

You can simplify it a bit more …

(1+logx)xxex^x

11. Jan 22, 2009

### Mentallic

Re: 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:

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

There is growing interest in the field of BS-mathematics

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

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