Производная (cot(x))^cos(x)

Вы ввели

   cos(x)   
cot      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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