Производная (tanh(x))^(1/x)

Вы ввели

x _________
\/ tanh(x) 

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

             /                                                                                                                                                                                                   /                                 2                                     \                    \ 
             |                              3                                                                                                                                     /                        2   \ |                  /         2   \                       /         2   \|                    | 
             |/                        2   \                                                                                                                                      |log(tanh(x))   -1 + tanh (x)| |           2      \-1 + tanh (x)/    2*log(tanh(x))   2*\-1 + tanh (x)/|                    | 
             ||log(tanh(x))   -1 + tanh (x)|                     2                    3                                                                                     2   3*|------------ + -------------|*|-2 + 2*tanh (x) - ---------------- + -------------- + -----------------|                    | 
             ||------------ + -------------|      /         2   \      /         2   \                                  /         2   \                      /         2   \      \     x            tanh(x)   / |                          2                 2             x*tanh(x)    |     /         2   \| 
 x _________ |\     x            tanh(x)   /    4*\-1 + tanh (x)/    2*\-1 + tanh (x)/      /         2   \           6*\-1 + tanh (x)/   6*log(tanh(x))   3*\-1 + tanh (x)/                                     \                      tanh (x)             x                           /   6*\-1 + tanh (x)/| 
-\/ tanh(x) *|------------------------------- - ------------------ + ------------------ + 4*\-1 + tanh (x)/*tanh(x) + ----------------- + -------------- - ------------------ + ---------------------------------------------------------------------------------------------------------- + -----------------| 
             |                2                      tanh(x)                  3                                               x                  3                   2                                                              x                                                             2           | 
             \               x                                            tanh (x)                                                              x              x*tanh (x)                                                                                                                        x *tanh(x)   / 
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                       x                                                                                                                                                        

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