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Please help me calculate the first derivate of the function $f(x)=x^{x^{x}}$. Thanks.

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    Is $f(x)=x^{(x^x)}$ or $f(x)=(x^x)^x$?2012-09-26

2 Answers 2

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Implement the formula:

1) $\log_a x^n=n\log_a x$

2) $\ln f(x)=\frac{1}{f(x)}\cdot f'(x)$

3) $[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)$

4) $[\ln x]'=\frac{1}{x}$

5) $a^{x_1}:a^{x_2}=\frac{a^{x_{1}}}{a^{x_{2}}}=a^{x_1-x_2}$

6) $x^{-n}=\frac{1}{x^n}$

Now turn find the derivate of this function.

$f(x)=x^{x^{x}}$/$\cdot\ln$

$\ln f(x)=\ln x^{x^{x}}$

$\ln f(x)=x^x\ln x$ / $^'$

$[\ln f(x)]'=(x^x\ln x)'$

$\frac{1}{f(x)}\cdot f'(x)=(x^x\ln x)'$

$\frac{1}{f(x)}\cdot f'(x)=(x^x)'\ln x + x^x(\ln x)'$

$\frac{1}{f(x)}\cdot f'(x)=(x^x)'\ln x + x^x\cdot\frac{1}{x}$

$\frac{1}{f(x)}\cdot f'(x)=(x^x)'\ln x + x^x\cdot x^{-1}$

$\frac{1}{f(x)}\cdot f'(x)=(x^x)'\ln x + x^{x-1}$ / $\cdot$ $f(x)$

$f(x)\cdot\frac{1}{f(x)}\cdot f'(x)=f(x)[(x^x)'\ln x + x^{x-1}]$

$f'(x)=f(x)[(x^x)'\ln x + x^{x-1}]$

$f'(x)=f(x)[f_1(x)\ln x + x^{x-1}]$

Now find the derivate of the function $f_1(x)=x^x$.

$f_1(x)=x^x$/$\cdot\ln$

$\ln f_1(x)=\ln x^x$

$\ln f_1(x)=x\ln x$/'

$[\ln f_1(x)]'=[x\ln x]'$

$\frac{1}{f_1(x)}\cdot f'_1(x)=x'\ln x+x(\ln x)'$

$\frac{1}{f_1(x)}\cdot f'_1(x)=\ln x+x\cdot\frac{1}{x}$

$\frac{1}{f_1(x)}\cdot f'_1(x)=\ln x+1$/$\cdot$ $f_1(x)$

$f'_1(x)=f_1(x)(\ln x+1)$

$f'_1(x)=(x^x)'=x^x(\ln x+1)$

Definitly:

$f'(x)=(x^{x^{x}})'=f(x)[f_1(x)\ln x + x^{x-1}]$

$f'(x)=(x^{x^{x}})'=x^{x^{x}}[x^x(\ln x+1)\ln x + x^{x-1}]$

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    It is "D-E-F-I-N-I-T-E-L-Y". It means, losely, "for sure". You might want to be using "Finally" or "All things considered" instead.2012-09-26
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Hint:

Try calculating $\log{f(x)}$