Производная sin(3*x)^x

Вы ввели

   x     
sin (3*x)

$$\sin^{x}{\left (3 x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

          /                              2                             2     \
   x      |/3*x*cos(3*x)                \          6*cos(3*x)   9*x*cos (3*x)|
sin (3*x)*||------------ + log(sin(3*x))|  - 9*x + ---------- - -------------|
          |\  sin(3*x)                  /           sin(3*x)         2       |
          \                                                       sin (3*x)  /

$$\left(- 9 x - \frac{9 x \cos^{2}{\left (3 x \right )}}{\sin^{2}{\left (3 x \right )}} + \left(\frac{3 x \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}} + \log{\left (\sin{\left (3 x \right )} \right )}\right)^{2} + \frac{6 \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}}\right) \sin^{x}{\left (3 x \right )}$$

Третья производная

          /                                    3         2                                         /                          2     \           3                     \
   x      |      /3*x*cos(3*x)                \    27*cos (3*x)     /3*x*cos(3*x)                \ |      2*cos(3*x)   3*x*cos (3*x)|   54*x*cos (3*x)   54*x*cos(3*x)|
sin (3*x)*|-27 + |------------ + log(sin(3*x))|  - ------------ - 9*|------------ + log(sin(3*x))|*|3*x - ---------- + -------------| + -------------- + -------------|
          |      \  sin(3*x)                  /        2            \  sin(3*x)                  / |       sin(3*x)         2       |        3              sin(3*x)  |
          \                                         sin (3*x)                                      \                     sin (3*x)  /     sin (3*x)                   /

$$\left(\frac{54 x \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}} + \frac{54 x \cos^{3}{\left (3 x \right )}}{\sin^{3}{\left (3 x \right )}} + \left(\frac{3 x \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}} + \log{\left (\sin{\left (3 x \right )} \right )}\right)^{3} - 9 \left(\frac{3 x \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}} + \log{\left (\sin{\left (3 x \right )} \right )}\right) \left(3 x + \frac{3 x \cos^{2}{\left (3 x \right )}}{\sin^{2}{\left (3 x \right )}} - \frac{2 \cos{\left (3 x \right )}}{\sin{\left (3 x \right )}}\right) - 27 - \frac{27 \cos^{2}{\left (3 x \right )}}{\sin^{2}{\left (3 x \right )}}\right) \sin^{x}{\left (3 x \right )}$$