Производная x^(cot(x))

Вы ввели

 cot(x)
x      

$$x^{\cot{\left (x \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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