Differentiate y = x^x^x^x^...^x - Clues for Substitution

[Reshma]

Differentiate this?

I need some help to differentiate this function:
y = x^x^x^x^...^x
I am sure there's got to be some appropriate substitution for the x^ term. Any clues?

 

[Curious3141]

I need some help to differentiate this function:
y = x^x^x^x^...^x
I am sure there's got to be some appropriate substitution for the x^ term. Any clues?

I'm assuming you meant the hyperpower function, which is the infinite power tower function. The x's go "all the way up".

You should read that function as $$y = x^{x^{x^{x^...}}}$$, that is, evaluate from the top down.

Then you can reexpress that as $$y = x^y$$

Take natural logs of both sides and differentiate implicitly.

$$\ln{y} = y\ln{x}$$

$$\frac{y'}{y} = \frac{y}{x} + y'\ln{x}$$

Group the terms together.

$$y'(\frac{1}{y} - \ln{x}) = \frac{y}{x}$$

And you can carry out further simplification yourself.

 

[benorin]

The given function is a so-called power tower.
It looks that the given function had finitely many levels, though.

You might try defining

$$f:y\rightarrow y^{x}$$,

and use f of f of ... of f and chain rule.

 

[Reshma]

Thanks Curious4131 and benorin!