Производная (cos(x))^(1/x)

Вы ввели

x ________
\/ cos(x) 

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

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

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

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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                /

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

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                                       

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

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                                                                                     

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

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