Производная x^x^x^x

Вы ввели

 / / x\\
 | \x /|
 \x    /
x       

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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