Производная x^x^x

Вы ввели

 / x\
 \x /
x    

$$x^{x^{x}}$$

  / / x\\
d | \x /|
--\x    /
dx       

$$\frac{d}{d x} x^{x^{x}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$

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Ответ:

$\left(\log{\left(x^{x} \right)} + 1\right) \left(x^{x}\right)^{x^{x}}$

Первая производная

 / x\ / x                         \
 \x / |x     x                    |
x    *|-- + x *(1 + log(x))*log(x)|
      \x                          /

$$x^{x^{x}} \left(x^{x} \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{x^{x}}{x}\right)$$

Вторая производная

    / x\ /                                            2                                        \
 x  \x / |  1    log(x)    x /1                      \                2          2*(1 + log(x))|
x *x    *|- -- + ------ + x *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) + --------------|
         |   2     x         \x                      /                                 x       |
         \  x                                                                                  /

$$x^{x} x^{x^{x}} \left(x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} + \left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right)$$

Третья производная

    / x\ /                                        3                                                                  2                                                                                                                 \
 x  \x / |2    3     2*x /1                      \                3          log(x)   3*(1 + log(x))   3*(1 + log(x))    3*(1 + log(x))*log(x)      x /1                      \ /  1    log(x)               2          2*(1 + log(x))\|
x *x    *|-- + -- + x   *|- + (1 + log(x))*log(x)|  + (1 + log(x)) *log(x) - ------ - -------------- + --------------- + --------------------- + 3*x *|- + (1 + log(x))*log(x)|*|- -- + ------ + (1 + log(x)) *log(x) + --------------||
         | 3    2        \x                      /                              2            2                x                    x                  \x                      / |   2     x                                   x       ||
         \x    x                                                               x            x                                                                                   \  x                                                  //

$$x^{x} x^{x^{x}} \left(x^{2 x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{3} + 3 x^{x} \left(\left(\log{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\log{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + \frac{2 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{\log{\left(x \right)}}{x} - \frac{1}{x^{2}}\right) + \left(\log{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{x} + \frac{3 \left(\log{\left(x \right)} + 1\right) \log{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} + 1\right)}{x^{2}} - \frac{\log{\left(x \right)}}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right)$$

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