Производная (log(x))^2^x

Вы ввели

        / x\
        \2 /
(log(x))    

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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