Results 1 to 2 of 2

Thread: stuck on second x^x

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    124

    stuck on second x^x

    $\displaystyle \begin{gathered}
    f(x) = {x^{{x^x}}} \hfill \\
    \ln y = {x^x}\ln x \hfill \\
    \frac{1}
    {y}y' = \frac{1}
    {x}*\frac{1}
    {x} \hfill \\
    \end{gathered}
    $

    but for the x^x, ln is needed but is
    $\displaystyle \ln \ln y = \ln ({x^x}\ln x)
    $

    okay?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    849
    Hello, genlovesmusic09!

    You are correct . . .


    Differentiate: .$\displaystyle y \:=\: x^{x^x}$

    Take logs: .$\displaystyle \ln(y) \;=\;\ln\left(x^{x^x}\right) \quad\Rightarrow\quad \ln(y)\;=\;x^x\cdot\ln(x) $

    Take logs: .$\displaystyle \ln\left[\ln(y)\right] \;=\;\ln\left[x^x\cdot\ln(x)\right] \;=\; \ln\left[x^x\right] + \ln\left[\ln(x)\right]$

    And we have: .$\displaystyle \ln\left[\ln(y)\right] \;=\;x\cdot\ln(x) + \ln\left[\ln(x)\right] $


    Differentiate: .$\displaystyle \frac{1}{\ln(y)}\!\cdot\!\frac{1}{y}\!\cdot\! y' \;\;=\;\;x\!\cdot\!\frac{1}{x} \;+\; 1\!\cdot\!\ln(x) \;+\; \frac{1}{\ln(x)}\!\cdot\!\frac{1}{x} $

    . . . . . . . . . . . . $\displaystyle \frac{y'}{y\!\cdot\!\ln(y)} \;=\;1 + \ln(x) + \frac{1}{x\!\cdot\!\ln(x)} $

    . . . . . . . . . . . . . . . $\displaystyle y' \;=\;y \cdot \ln(y)\cdot\bigg[1 + \ln(x) + \frac{1}{x\!\cdot\!\ln(x)} \bigg] $

    . . . . . . . . . . . . . . . $\displaystyle y' \;=\;x^{x^x}\cdot\ln\left(x^{x^x}\right)\cdot\bigg[1 + \ln(x) + \frac{1}{x\!\cdot\!\ln(x)} \bigg] $

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Stuck
    Posted in the Algebra Forum
    Replies: 6
    Last Post: Jan 21st 2010, 11:09 AM
  2. stuck with sin^4(x)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 24th 2009, 04:56 PM
  3. Stuck on a Log :(
    Posted in the Algebra Forum
    Replies: 9
    Last Post: Jan 21st 2008, 12:10 PM
  4. somebody help me plz im so stuck
    Posted in the Math Topics Forum
    Replies: 4
    Last Post: Sep 30th 2007, 11:43 AM
  5. Stuck, Stuck, Stuck - Need Help Urgently
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 17th 2005, 05:46 AM

Search Tags


/mathhelpforum @mathhelpforum