Производная sin(sin(x))^(log(x))

Вы ввели

   log(x)        
sin      (sin(x))

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

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

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

Вторая производная

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

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

Третья производная

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

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