Производная (acos(x))^sin(x)

Вы ввели

    sin(x)   
acos      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

    sin(x)    /                             sin(x)      \
acos      (x)*|cos(x)*log(acos(x)) - -------------------|
              |                         ________        |
              |                        /      2         |
              \                      \/  1 - x  *acos(x)/

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

Вторая производная

              /                                           2                                                                                       \
    sin(x)    |/                             sin(x)      \                                sin(x)               2*cos(x)              x*sin(x)     |
acos      (x)*||cos(x)*log(acos(x)) - -------------------|  - log(acos(x))*sin(x) + ------------------ - ------------------- - -------------------|
              ||                         ________        |                          /      2\     2         ________                   3/2        |
              ||                        /      2         |                          \-1 + x /*acos (x)     /      2            /     2\           |
              \\                      \/  1 - x  *acos(x)/                                               \/  1 - x  *acos(x)   \1 - x /   *acos(x)/

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

Третья производная

              /                                           3                                                                                                                                                                                                                                                                                                           2           \
    sin(x)    |/                             sin(x)      \                            /                             sin(x)      \ /                            sin(x)               2*cos(x)              x*sin(x)     \          sin(x)               2*sin(x)               3*sin(x)             3*cos(x)             3*x*cos(x)            3*x*sin(x)           3*x *sin(x)    |
acos      (x)*||cos(x)*log(acos(x)) - -------------------|  - cos(x)*log(acos(x)) - 3*|cos(x)*log(acos(x)) - -------------------|*|log(acos(x))*sin(x) - ------------------ + ------------------- + -------------------| - ------------------- - -------------------- + ------------------- + ------------------ - ------------------- - ------------------- - -------------------|
              ||                         ________        |                            |                         ________        | |                      /      2\     2         ________                   3/2        |           3/2                   3/2               ________           /      2\     2              3/2                    2                    5/2        |
              ||                        /      2         |                            |                        /      2         | |                      \-1 + x /*acos (x)     /      2            /     2\           |   /     2\              /     2\        3        /      2            \-1 + x /*acos (x)   /     2\              /      2\      2      /     2\           |
              \\                      \/  1 - x  *acos(x)/                            \                      \/  1 - x  *acos(x)/ \                                           \/  1 - x  *acos(x)   \1 - x /   *acos(x)/   \1 - x /   *acos(x)   \1 - x /   *acos (x)   \/  1 - x  *acos(x)                        \1 - x /   *acos(x)   \-1 + x / *acos (x)   \1 - x /   *acos(x)/

$$\left(- \frac{3 x^{2} \sin{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{5}{2}} \operatorname{acos}{\left (x \right )}} - \frac{3 x \sin{\left (x \right )}}{\left(x^{2} - 1\right)^{2} \operatorname{acos}^{2}{\left (x \right )}} - \frac{3 x \cos{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} + \left(\log{\left (\operatorname{acos}{\left (x \right )} \right )} \cos{\left (x \right )} - \frac{\sin{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}\right)^{3} - 3 \left(\log{\left (\operatorname{acos}{\left (x \right )} \right )} \cos{\left (x \right )} - \frac{\sin{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}\right) \left(\frac{x \sin{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} + \log{\left (\operatorname{acos}{\left (x \right )} \right )} \sin{\left (x \right )} - \frac{\sin{\left (x \right )}}{\left(x^{2} - 1\right) \operatorname{acos}^{2}{\left (x \right )}} + \frac{2 \cos{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}\right) - \log{\left (\operatorname{acos}{\left (x \right )} \right )} \cos{\left (x \right )} + \frac{3 \cos{\left (x \right )}}{\left(x^{2} - 1\right) \operatorname{acos}^{2}{\left (x \right )}} + \frac{3 \sin{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}} - \frac{\sin{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} - \frac{2 \sin{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}^{3}{\left (x \right )}}\right) \operatorname{acos}^{\sin{\left (x \right )}}{\left (x \right )}$$