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Math Help - derivatives

  1. #1
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    derivatives

    Hei, guys! I wonder you can help me. here is my question: i must find derivative of x^x^x^x^x... infinitive. f`(x)=(x^x^x^x...)` .
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  2. #2
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    Re: derivatives

    Hello, thealivision!

    \text{Differentiate: }\:y \;=\;x^{x^{x^\cdots}}

    \text{Take logs: }\:\ln(y) \;=\;\ln\left(x^{x^{x^\cdots}}\right) \quad\Rightarrow\quad \ln(y) \;=\;\underbrace{x^{x^{x^\cdots}}}}_{\text{This is }y}\ln x

    \text{We have: }\:\ln(y) \;=\;y\ln(x)

    Differentiate implicitly: . \frac{1}{y}y' \;=\;y\,\frac{1}{x} + y'\ln(x) \quad\Rightarrow\quad \frac{y'}{y} - y'\ln(x) \;=\;\frac{y}{x}

    Multiply by xy\!:\;\;xy' - xyy'\ln(x) \:=\:y^2

    Factor: . x\big[1-y\ln(x)\big]y' \;=\;y^2

    Therefore: . y' \;=\;\frac{y^2}{x\big[1-y\ln(x)\big]}
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  3. #3
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    Re: derivatives

    thanks a lot
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