Производная cos(x)^(log(x))

Вы ввели

   log(x)   
cos      (x)

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

d /   log(x)   \
--\cos      (x)/
dx              

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

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)               /

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

Третья производная

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

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

График