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

  • Thread starter Reshma
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In summary, the given function is a power tower, also known as a hyperpower function. To differentiate it, you can reexpress it as y = x^y and use implicit differentiation, resulting in the equation y'(\frac{1}{y} - \ln{x}) = \frac{y}{x}. Further simplification can be carried out by defining a function f:y\rightarrow y^{x} and using the chain rule.
  • #1
Reshma
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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?
 
Last edited:
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  • #2
Reshma said:
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 [tex]y = x^{x^{x^{x^...}}}[/tex], that is, evaluate from the top down.

Then you can reexpress that as [tex]y = x^y[/tex]

Take natural logs of both sides and differentiate implicitly.

[tex]\ln{y} = y\ln{x}[/tex]

[tex]\frac{y'}{y} = \frac{y}{x} + y'\ln{x}[/tex]

Group the terms together.

[tex]y'(\frac{1}{y} - \ln{x}) = \frac{y}{x}[/tex]

And you can carry out further simplification yourself.
 
  • #3
The given function is a so-called power tower.
It looks that the given function had finitely many levels, though.

You might try defining

[tex]f:y\rightarrow y^{x}[/tex],

and use f of f of ... of f and chain rule.
 
Last edited:
  • #4
Thanks Curious4131 and benorin!
 

1. What is the first step in solving this equation?

The first step in solving this equation is to identify the pattern and determine the number of iterations of the x^x operation. This will help in determining the number of layers of substitution that need to be done.

2. How do I perform substitution in this equation?

To perform substitution, replace the variable x with the given value or expression. Then, simplify the equation using the appropriate exponent rules.

3. Can I use any value for x in this equation?

No, the value of x must be a positive real number for this equation to be valid. Negative numbers or zero would result in undefined values.

4. Is there a shortcut to solving this equation?

There is no shortcut to solving this equation. Each layer of substitution must be done individually and the result must be evaluated before moving on to the next layer.

5. How do I know when to stop substituting and evaluate the final expression?

You should continue substituting until there are no more exponents left to simplify. The final expression will be a number that can be evaluated using a calculator or by hand.

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