Производная cot(x)^acot(x)

Вы ввели

   acot(x)   
cot       (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

               /                                     3                                                           /                                       2                                            \                                    2                          2                  3                                                                                \
               |/              /       2   \        \                      /              /       2   \        \ |                          /       2   \                                /       2   \|     /       2   \     /       2   \              /       2   \      /       2   \                                                2                   /       2   \|
    acot(x)    ||log(cot(x))   \1 + cot (x)/*acot(x)|    2*log(cot(x))     |log(cot(x))   \1 + cot (x)/*acot(x)| |  /       2   \           \1 + cot (x)/ *acot(x)   2*x*log(cot(x))   2*\1 + cot (x)/|   6*\1 + cot (x)/   4*\1 + cot (x)/ *acot(x)   3*\1 + cot (x)/    2*\1 + cot (x)/ *acot(x)     /       2   \                  8*x *log(cot(x))   6*x*\1 + cot (x)/|
-cot       (x)*||----------- + ---------------------|  - ------------- + 3*|----------- + ---------------------|*|2*\1 + cot (x)/*acot(x) - ---------------------- + --------------- + ---------------| + --------------- - ------------------------ - ---------------- + ------------------------ + 4*\1 + cot (x)/*acot(x)*cot(x) + ---------------- + -----------------|
               ||        2             cot(x)       |              2       |        2             cot(x)       | |                                    2                         2      /     2\       |             2                cot(x)            /     2\    2                 3                                                           3                2       |
               |\   1 + x                           /      /     2\        \   1 + x                           / |                                 cot (x)              /     2\       \1 + x /*cot(x)|        1 + x                                   \1 + x /*cot (x)           cot (x)                                                /     2\         /     2\        |
               \                                           \1 + x /                                              \                                                      \1 + x /                      /                                                                                                                                  \1 + x /         \1 + x / *cot(x)/

$$- \left(\frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} \log{\left (\cot{\left (x \right )} \right )} + \frac{6 x \left(\cot^{2}{\left (x \right )} + 1\right)}{\left(x^{2} + 1\right)^{2} \cot{\left (x \right )}} + \left(\frac{\operatorname{acot}{\left (x \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) + \frac{1}{x^{2} + 1} \log{\left (\cot{\left (x \right )} \right )}\right)^{3} + 3 \left(\frac{\operatorname{acot}{\left (x \right )}}{\cot{\left (x \right )}} \left(\cot^{2}{\left (x \right )} + 1\right) + \frac{1}{x^{2} + 1} \log{\left (\cot{\left (x \right )} \right )}\right) \left(\frac{2 x}{\left(x^{2} + 1\right)^{2}} \log{\left (\cot{\left (x \right )} \right )} - \frac{\left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot^{2}{\left (x \right )}} \operatorname{acot}{\left (x \right )} + 2 \left(\cot^{2}{\left (x \right )} + 1\right) \operatorname{acot}{\left (x \right )} + \frac{2 \cot^{2}{\left (x \right )} + 2}{\left(x^{2} + 1\right) \cot{\left (x \right )}}\right) + \frac{2 \left(\cot^{2}{\left (x \right )} + 1\right)^{3}}{\cot^{3}{\left (x \right )}} \operatorname{acot}{\left (x \right )} - \frac{4 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\cot{\left (x \right )}} \operatorname{acot}{\left (x \right )} + 4 \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} \operatorname{acot}{\left (x \right )} - \frac{3 \left(\cot^{2}{\left (x \right )} + 1\right)^{2}}{\left(x^{2} + 1\right) \cot^{2}{\left (x \right )}} + \frac{1}{x^{2} + 1} \left(6 \cot^{2}{\left (x \right )} + 6\right) - \frac{2}{\left(x^{2} + 1\right)^{2}} \log{\left (\cot{\left (x \right )} \right )}\right) \cot^{\operatorname{acot}{\left (x \right )}}{\left (x \right )}$$