Производная (acot(x))^x-1

Вы ввели

    x       
acot (x) - 1

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

Подробное решение

1. дифференцируем $\operatorname{acot}^{x}{\left (x \right )} - 1$ почленно:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   2. Производная постоянной $-1$ равна нулю.

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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