Производная log(x)^(sin(x))

Вы ввели

   sin(x)   
log      (x)

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

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

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

Вторая производная

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

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

Третья производная

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

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