Производная (cos(x))^acot(x)

Вы ввели

   acot(x)   
cos       (x)

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

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

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

Вторая производная

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

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

Третья производная

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

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