Производная x^(sin(x))

Вы ввели

 sin(x)
x      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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

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

Вторая производная

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

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

Третья производная

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

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