Производная (log(x))^(cos(x))

Вы ввели

   cos(x)   
log      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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