Производная cos(x)^(1/x)

Вы ввели

x ________
\/ cos(x) 

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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