Производная (cos(x+2))^log(x)

Вы ввели

   log(x)       
cos      (x + 2)

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

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

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

Вторая производная

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

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

Третья производная

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

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