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    hei, guys! I really enjoy this website and i wonder if you can help me with a problem. I know this was given to imperial last year and i didn`t know exactly the answer. f`(x^x^x^x...) infinitive.i found lots of wrong solutions of this problem. pretty please help me. Thanks. Andreea
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    (Original post by alistarandreea)
    hei, guys! I really enjoy this website and i wonder if you can help me with a problem. I know this was given to imperial last year and i didn`t know exactly the answer. f`(x^x^x^x...) infinitive.i found lots of wrong solutions of this problem. pretty please help me. Thanks. Andreea
    is this possible? is it even possible to sketch the graph?

    for some idea of what the derivative of a similar function is http://www.wolframalpha.com/input/?i=x^x^x^x^x^x^x^x^x+ maybe that will help in some way?
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    (Original post by alistarandreea)
    ...
    Question is unclear, but I'm going to assume you want the derivative of y=x^{x^{x^{x^{\,\cdots}}}}

    Noting that y=x^y:

    y'=\left(y'\ln x+\dfrac{y}{x}\right)x^y\iff y'=\dfrac{y x^{y-1}}{1-x^y\ln x}

    Which we can rewrite as y'=\dfrac{1}{x}\left(x^{x^{x^{x^  {\,\cdots}}}}\right)^2 \dfrac{1}{1-\ln \left(x^{x^{x^{x^{\,\cdots}}}} \right)}


    This is not the end of the story: y is only defined within a certain range: \frac{1}{e^e} \leq x\leq e^{\frac{1}{e}}

    You may want to think about why this is, and prove it (the upper bound is relatively easy, the lower much less so)
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    Thanks a lot. So i must prove and that`s all? That's a sly trick by redefining y as y=x^y and I`m not sure is correct.
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    (Original post by alistarandreea)
    Thanks a lot. So i must prove and that`s all? That's a sly trick by redefining y as y=x^y and I`m not sure is correct.
    The trick is valid, a better way to look at it is to take the natural log of each side.
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    (Original post by alistarandreea)
    Thanks a lot. So i must prove and that`s all? That's a sly trick by redefining y as y=x^y and I'm not sure is correct.
    It's perfectly valid. No you don't have to prove that x must lie in that range; that suggestion was your own interest rather than anything else. I suppose it also depends on how rigorous you want to be. Technically when you define a function/take a derivative, you should specify the domain over which the manipulations are valid. Warning though: if you had trouble finding the derivative, this is on a whole new level of difficulty.
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    ok. i gave to my math teacher this answer to the problem and he said me isn`t correct. firstly, i began with ln(y)=ln (x) * y and he said isn`t correct so i stopped to do smth. that`s why i asked you. but thanks anyway have a great day
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    (Original post by alistarandreea)
    ok. i gave to my math teacher this answer to the problem and he said me isn`t correct.
    Either you've mis-stated the question or your teacher is wrong.
 
 
 
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