Производная x^acot(x)

Вы ввели

 acot(x)
x       

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

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

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

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

Вторая производная

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

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

Третья производная

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

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