Производная x^cot(x)

Вы ввели

 cot(x)
x      

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

d / cot(x)\
--\x      /
dx         

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

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                           /

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

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                                                     /

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

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                                                     /                                                          /

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

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