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Thread: seemingly simple derivative

  1. #1
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    seemingly simple derivative

    hello all. my friend has asked me for help on a derivation but i've come up blank.

    the derivation in question is:

    $\displaystyle \frac{d}{dx}lnx^{lnx}$

    i thought $\displaystyle lnx$

    he thought $\displaystyle 2\frac{lnx}{x}$

    what do YOU think?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Swampscott, MA
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    Quote Originally Posted by makieru View Post
    hello all. my friend has asked me for help on a derivation but i've come up blank.

    the derivation in question is:

    $\displaystyle \frac{d}{dx}lnx^{lnx}$

    i thought $\displaystyle lnx$

    he thought $\displaystyle 2\frac{lnx}{x}$

    what do YOU think?
    It's $\displaystyle \frac{2\ln x}{x}$. Here's why:

    $\displaystyle \ln\left(x^{\ln x}\right)=(\ln x)(\ln x)=(\ln x)^2$

    Use the chain rule:

    $\displaystyle 2\ln x\cdot\frac{1}{x}=\frac{2\ln x}{x}$
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