Производная sin(x)^log(x)

Вы ввели

   log(x)   
sin      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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