Производная log(x)^(1/x)

Вы ввели

x ________
\/ log(x) 

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

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

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

Вторая производная

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

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

Третья производная

           /                                                                      3     /    1                 \ /   1                        3   \\
           |                                              /    1                 \    3*|- ------ + log(log(x))|*|------- - 2*log(log(x)) + ------||
           |                                              |- ------ + log(log(x))|      \  log(x)              / |   2                      log(x)||
x ________ |                    2         6        11     \  log(x)              /                               \log (x)                         /|
\/ log(x) *|-6*log(log(x)) + ------- + ------- + ------ - ------------------------- + -------------------------------------------------------------|
           |                    3         2      log(x)                2                                            x                              |
           \                 log (x)   log (x)                        x                                                                            /
----------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          4                                                                         
                                                                         x                                                                          

$$\frac{1}{x^{4}} \left(- 6 \log{\left (\log{\left (x \right )} \right )} + \frac{11}{\log{\left (x \right )}} + \frac{6}{\log^{2}{\left (x \right )}} + \frac{2}{\log^{3}{\left (x \right )}} + \frac{3}{x} \left(\log{\left (\log{\left (x \right )} \right )} - \frac{1}{\log{\left (x \right )}}\right) \left(- 2 \log{\left (\log{\left (x \right )} \right )} + \frac{3}{\log{\left (x \right )}} + \frac{1}{\log^{2}{\left (x \right )}}\right) - \frac{1}{x^{2}} \left(\log{\left (\log{\left (x \right )} \right )} - \frac{1}{\log{\left (x \right )}}\right)^{3}\right) \log^{\frac{1}{x}}{\left (x \right )}$$