Производная cos(x)^x^-2

Вы ввели

        1 
        --
         2
        x 
(cos(x))  

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

        1                               
        --                              
         2                              
        x  /  2*log(cos(x))     sin(x) \
(cos(x))  *|- ------------- - ---------|
           |         3         2       |
           \        x         x *cos(x)/

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

Вторая производная

        1  /                             2                                     \
        -- |     /sin(x)   2*log(cos(x))\                                      |
         2 |     |------ + -------------|       2                              |
        x  |     \cos(x)         x      /    sin (x)   6*log(cos(x))   4*sin(x)|
(cos(x))  *|-1 + ------------------------- - ------- + ------------- + --------|
           |                  2                 2             2        x*cos(x)|
           \                 x               cos (x)         x                 /
--------------------------------------------------------------------------------
                                        2                                       
                                       x                                        

$$\frac{1}{x^{2}} \left(- \frac{\sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} - 1 + \frac{4 \sin{\left (x \right )}}{x \cos{\left (x \right )}} + \frac{1}{x^{2}} \left(\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} + \frac{2}{x} \log{\left (\cos{\left (x \right )} \right )}\right)^{2} + \frac{6}{x^{2}} \log{\left (\cos{\left (x \right )} \right )}\right) \cos^{\frac{1}{x^{2}}}{\left (x \right )}$$

Третья производная

           /                                                                                                               /       2                              \            \
        1  |                            3                                                         /sin(x)   2*log(cos(x))\ |    sin (x)   6*log(cos(x))   4*sin(x)|            |
        -- |    /sin(x)   2*log(cos(x))\                                                        3*|------ + -------------|*|1 + ------- - ------------- - --------|            |
         2 |    |------ + -------------|                                     3                    \cos(x)         x      / |       2             2        x*cos(x)|        2   |
        x  |6   \cos(x)         x      /    24*log(cos(x))   2*sin(x)   2*sin (x)   18*sin(x)                              \    cos (x)         x                 /   6*sin (x)|
(cos(x))  *|- - ------------------------- - -------------- - -------- - --------- - --------- + ------------------------------------------------------------------- + ---------|
           |x                4                     3          cos(x)        3        2                                            2                                        2   |
           \                x                     x                      cos (x)    x *cos(x)                                    x                                    x*cos (x)/
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                        2                                                                                       
                                                                                       x                                                                                        

$$\frac{1}{x^{2}} \left(- \frac{2 \sin^{3}{\left (x \right )}}{\cos^{3}{\left (x \right )}} - \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )}} + \frac{6 \sin^{2}{\left (x \right )}}{x \cos^{2}{\left (x \right )}} + \frac{6}{x} + \frac{3}{x^{2}} \left(\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} + \frac{2}{x} \log{\left (\cos{\left (x \right )} \right )}\right) \left(\frac{\sin^{2}{\left (x \right )}}{\cos^{2}{\left (x \right )}} + 1 - \frac{4 \sin{\left (x \right )}}{x \cos{\left (x \right )}} - \frac{6}{x^{2}} \log{\left (\cos{\left (x \right )} \right )}\right) - \frac{18 \sin{\left (x \right )}}{x^{2} \cos{\left (x \right )}} - \frac{24}{x^{3}} \log{\left (\cos{\left (x \right )} \right )} - \frac{1}{x^{4}} \left(\frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} + \frac{2}{x} \log{\left (\cos{\left (x \right )} \right )}\right)^{3}\right) \cos^{\frac{1}{x^{2}}}{\left (x \right )}$$