Производная sin(x^cos(x))

Вы ввели

   / cos(x)\
sin\x      /

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

Подробное решение

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

2. Производная синуса есть косинус:

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

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

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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