Производная (acot(x))^(log(x))

Вы ввели

    log(x)   
acot      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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