Производная sin(x)^acot(x)

Вы ввели

   acot(x)   
sin       (x)

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

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

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

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

Вторая производная

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

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

Третья производная

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

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