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How can we calculate the value of $ \frac{dy}{dx} (log (x))^{x}$

I tried doing it the following way :

Let $ y= (\log (x))^{x} $

$ \log y = x \log \log (x)$

Then differentiating both sides with respect to $x$ but its not working.

Mojo Jojo
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  • It should work. – kmitov Mar 02 '16 at 06:04
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    Saying it's not working doesn't give any information, so it's not really possible to answer your question. That's a valid approach, but we don't know where you're having trouble implementing it. –  Mar 02 '16 at 06:08
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    Do you mean $\frac{d}{dx}\left((\log (x))^x\right)$? – mathlove Mar 02 '16 at 06:09
  • this type of derivative work out in a convenient manner as following: $ d(\ln y)/dx = (dy/dx) / y$ hence $dy/dx = y * d(\ln y)/dx$. In your case $\ln y = x \log \log x$, which one can differentiate as usual. – Chip Mar 02 '16 at 06:13
  • @kmitov It worked. I was making a mistake in applying the chain rule in last step. Thanks. – Mojo Jojo Mar 02 '16 at 06:25

1 Answers1

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$\log y=x\log \log x$ Differentiate both sides with respect to $x$ you get

$y'/y=\log \log x + x \frac{1}{\log x} \frac{1}{x}$

kmitov
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