Производная (acot(x))^x^2

Вы ввели

         / 2\
         \x /
(acot(x))    

$$\operatorname{acot}^{x^{2}}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

               /                                             /        2          4              2                     3                         \                                                                                                                        \
               |                                             |     7*x        4*x              x                   3*x                3*x       |                                                                                                                        |
               |                                           2*|3 - ------ + --------- + ------------------ - ----------------- + ----------------|                                                                                                                        |
               |                                             |         2           2           2                    2           /     2\        |                                                                                                                        |
          / 2\ |                                       3     |    1 + x    /     2\    /     2\      2      /     2\            \1 + x /*acot(x)|                                            /                          2                                        3      \|
          \x / | 3 /                         x        \      \             \1 + x /    \1 + x / *acot (x)   \1 + x / *acot(x)                   /       /                         x        \ |                         x                  4*x                 2*x       ||
-(acot(x))    *|x *|-2*log(acot(x)) + ----------------|  + -------------------------------------------------------------------------------------- + 3*x*|-2*log(acot(x)) + ----------------|*|2*log(acot(x)) - ------------------ - ---------------- + -----------------||
               |   |                  /     2\        |                                       /     2\                                                  |                  /     2\        | |                         2            /     2\                   2        ||
               |   \                  \1 + x /*acot(x)/                                       \1 + x /*acot(x)                                          \                  \1 + x /*acot(x)/ |                 /     2\      2      \1 + x /*acot(x)   /     2\         ||
               \                                                                                                                                                                             \                 \1 + x / *acot (x)                      \1 + x / *acot(x)//

$$- \left(x^{3} \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{acot}{\left (x \right )}} - 2 \log{\left (\operatorname{acot}{\left (x \right )} \right )}\right)^{3} + 3 x \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{acot}{\left (x \right )}} - 2 \log{\left (\operatorname{acot}{\left (x \right )} \right )}\right) \left(\frac{2 x^{3}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left (x \right )}} - \frac{x^{2}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}^{2}{\left (x \right )}} - \frac{4 x}{\left(x^{2} + 1\right) \operatorname{acot}{\left (x \right )}} + 2 \log{\left (\operatorname{acot}{\left (x \right )} \right )}\right) + \frac{1}{\left(x^{2} + 1\right) \operatorname{acot}{\left (x \right )}} \left(\frac{8 x^{4}}{\left(x^{2} + 1\right)^{2}} - \frac{6 x^{3}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}{\left (x \right )}} - \frac{14 x^{2}}{x^{2} + 1} + \frac{2 x^{2}}{\left(x^{2} + 1\right)^{2} \operatorname{acot}^{2}{\left (x \right )}} + \frac{6 x}{\left(x^{2} + 1\right) \operatorname{acot}{\left (x \right )}} + 6\right)\right) \operatorname{acot}^{x^{2}}{\left (x \right )}$$