Производная (cos(x))^log(x)

Вы ввели

   log(x)   
cos      (x)

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

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

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

Вторая производная

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

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