Производная sin(x^cot(x))

Вы ввели

   / cot(x)\
sin\x      /

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

Подробное решение

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

2. Производная синуса есть косинус:

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

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

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

        //                 2                       /       2   \                                      /       2   \       \                                               3                                                         3                                                         3                                                  /           /       2   \                                \                                                          /           /       2   \                                \             \
 cot(x) ||    /       2   \           2*cot(x)   3*\1 + cot (x)/        2    /       2   \          6*\1 + cot (x)/*cot(x)|    / cot(x)\   //       2   \          cot(x)\     / cot(x)\    2*cot(x) //       2   \          cot(x)\     / cot(x)\      cot(x) //       2   \          cot(x)\     / cot(x)\     //       2   \          cot(x)\ |cot(x)   2*\1 + cot (x)/     /       2   \              |    / cot(x)\      cot(x) //       2   \          cot(x)\ |cot(x)   2*\1 + cot (x)/     /       2   \              |    / cot(x)\|
x      *||- 2*\1 + cot (x)/ *log(x) + -------- + --------------- - 4*cot (x)*\1 + cot (x)/*log(x) + ----------------------|*cos\x      / - |\1 + cot (x)/*log(x) - ------| *cos\x      / + x        *|\1 + cot (x)/*log(x) - ------| *cos\x      / + 3*x      *|\1 + cot (x)/*log(x) - ------| *sin\x      / + 3*|\1 + cot (x)/*log(x) - ------|*|------ + --------------- - 2*\1 + cot (x)/*cot(x)*log(x)|*cos\x      / - 3*x      *|\1 + cot (x)/*log(x) - ------|*|------ + --------------- - 2*\1 + cot (x)/*cot(x)*log(x)|*sin\x      /|
        ||                                3              2                                                    x           |                \                         x   /                           \                         x   /                           \                         x   /                   \                         x   / |   2            x                                       |                          \                         x   / |   2            x                                       |             |
        \\                               x              x                                                                 /                                                                                                                                                                                                                      \  x                                                     /                                                          \  x                                                     /             /

$$x^{\cot{\left (x \right )}} \left(x^{2 \cot{\left (x \right )}} \left(\left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \cot{\left (x \right )}\right)^{3} \cos{\left (x^{\cot{\left (x \right )}} \right )} + 3 x^{\cot{\left (x \right )}} \left(\left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \cot{\left (x \right )}\right)^{3} \sin{\left (x^{\cot{\left (x \right )}} \right )} - 3 x^{\cot{\left (x \right )}} \left(\left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \cot{\left (x \right )}\right) \left(- 2 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} \cot{\left (x \right )} + \frac{1}{x} \left(2 \cot^{2}{\left (x \right )} + 2\right) + \frac{1}{x^{2}} \cot{\left (x \right )}\right) \sin{\left (x^{\cot{\left (x \right )}} \right )} - \left(\left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \cot{\left (x \right )}\right)^{3} \cos{\left (x^{\cot{\left (x \right )}} \right )} + 3 \left(\left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \cot{\left (x \right )}\right) \left(- 2 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} \cot{\left (x \right )} + \frac{1}{x} \left(2 \cot^{2}{\left (x \right )} + 2\right) + \frac{1}{x^{2}} \cot{\left (x \right )}\right) \cos{\left (x^{\cot{\left (x \right )}} \right )} + \left(- 2 \left(\cot^{2}{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} - 4 \left(\cot^{2}{\left (x \right )} + 1\right) \log{\left (x \right )} \cot^{2}{\left (x \right )} + \frac{6}{x} \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} + \frac{1}{x^{2}} \left(3 \cot^{2}{\left (x \right )} + 3\right) + \frac{2}{x^{3}} \cot{\left (x \right )}\right) \cos{\left (x^{\cot{\left (x \right )}} \right )}\right)$$