Производная cot(x)^sin(x)

Вы ввели

   sin(x)   
cot      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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