Производная (sin(x))^sin(x)

Вы ввели

   sin(x)   
sin      (x)

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

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)*(cos(x)*log(sin(x)) + cos(x))

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

Вторая производная

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

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

Третья производная

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

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