Производная cot(x)^tan(x)

Вы ввели

   tan(x)   
cot      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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