Производная sin(x^(sin(x)))

Вы ввели

   / sin(x)\
sin\x      /

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

Подробное решение

1. Заменим $u = x^{\sin{\left (x \right )}}$.

2. Производная синуса есть косинус:

   $\frac{d}{d u} \sin{\left (u \right )} = \cos{\left (u \right )}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} x^{\sin{\left (x \right )}}$:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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