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What is the derivative of x^(x^(x^(x^…)))?

This question was submitted by @NestorAiros via Twitter. Thank you, Nestor! You have certainly come to the right place for your math questions! Congratulations on being the very first user question!

Actual Question:
“@CalculusTricks hey bro! Idon’t speak lot of english but here is myquestion,hope u understand: how i do differiancibility: y=x^x^x .. y’=?”

Your english is good! Very understandable.

What is the derivative of ?

Remember that  .

Let’s take this on step at a time. Maybe we can find a pattern. First, simplify the problem. What is the derivative of ?

Use implicit differentiation.

Now, what about ?

That is as far as this one simplifies. The next couple are going to get much, much messier. But I’ll  do out one more completely.

Let’s find the derivative of !

See? Messy. This starts to get quite tedious, so I am just going to post the solutions to the next few steps.

It took me a bit to figure out what the pattern was, but there is one! Try writing all of them on paper, and you can see it much better. Then, you can imply the answers for any following steps in this problem. I really hope this answers your question. Please let me know either way by commenting below. If you have further questions, email me at admin@calculustricks.com. I will reply to you via email, and update this post if necessary.

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  1. NestorAiros says:

    Thanx a lot man!! There is a pattern but i can’t see it from the beggining, need more practice with the derivates, i’m starting my first six month period and i have.. Any dude i will post u on twitter and thanks a lot! This was really helpful!

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