Производная x^(x^(2*x))

Вы ввели

 / 2*x\
 \x   /
x      

$$x^{x^{2 x}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

 / 2*x\      /                                       2                                                     \
 \x   /  2*x |  1     2*x /1                        \    2*log(x)   4*(1 + log(x))                 2       |
x      *x   *|- -- + x   *|- + 2*(1 + log(x))*log(x)|  + -------- + -------------- + 4*(1 + log(x)) *log(x)|
             |   2        \x                        /       x             x                                |
             \  x                                                                                          /

$$x^{2 x} x^{x^{2 x}} \left(x^{2 x} \left(2 \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} + \frac{1}{x}\right)^{2} + 4 \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} + \frac{1}{x} \left(4 \log{\left (x \right )} + 4\right) + \frac{2}{x} \log{\left (x \right )} - \frac{1}{x^{2}}\right)$$

Третья производная

 / 2*x\      /                                          3                                                                       2                                                                                                                          \
 \x   /  2*x |2    6     4*x /1                        \    6*(1 + log(x))   2*log(x)                 3          12*(1 + log(x))       2*x /1                        \ /  1    2*log(x)   4*(1 + log(x))                 2       \   12*(1 + log(x))*log(x)|
x      *x   *|-- + -- + x   *|- + 2*(1 + log(x))*log(x)|  - -------------- - -------- + 8*(1 + log(x)) *log(x) + ---------------- + 3*x   *|- + 2*(1 + log(x))*log(x)|*|- -- + -------- + -------------- + 4*(1 + log(x)) *log(x)| + ----------------------|
             | 3    2        \x                        /           2             2                                      x                  \x                        / |   2      x             x                                |             x           |
             \x    x                                              x             x                                                                                      \  x                                                      /                         /

$$x^{2 x} x^{x^{2 x}} \left(x^{4 x} \left(2 \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} + \frac{1}{x}\right)^{3} + 3 x^{2 x} \left(2 \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} + \frac{1}{x}\right) \left(4 \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} + \frac{1}{x} \left(4 \log{\left (x \right )} + 4\right) + \frac{2}{x} \log{\left (x \right )} - \frac{1}{x^{2}}\right) + 8 \left(\log{\left (x \right )} + 1\right)^{3} \log{\left (x \right )} + \frac{12}{x} \left(\log{\left (x \right )} + 1\right)^{2} + \frac{12}{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x^{2}} \left(6 \log{\left (x \right )} + 6\right) - \frac{2}{x^{2}} \log{\left (x \right )} + \frac{6}{x^{2}} + \frac{2}{x^{3}}\right)$$