Производная x^(log(x)/x)

Вы ввели

 log(x)
 ------
   x   
x      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

 log(x)                                                                                                                                      
 ------ /                                                         3    3                                                                    \
   x    |                                            (-2 + log(x)) *log (x)   3*(-2 + log(x))*(2 - 3*log(x) + (-3 + 2*log(x))*log(x))*log(x)|
x      *|-12 + 11*log(x) - (-11 + 6*log(x))*log(x) - ---------------------- - --------------------------------------------------------------|
        |                                                       2                                           x                               |
        \                                                      x                                                                            /
---------------------------------------------------------------------------------------------------------------------------------------------
                                                                       4                                                                     
                                                                      x                                                                      

$$\frac{1}{x^{4}} x^{\frac{1}{x} \log{\left (x \right )}} \left(- \left(6 \log{\left (x \right )} - 11\right) \log{\left (x \right )} + 11 \log{\left (x \right )} - 12 - \frac{3}{x} \left(\log{\left (x \right )} - 2\right) \left(\left(2 \log{\left (x \right )} - 3\right) \log{\left (x \right )} - 3 \log{\left (x \right )} + 2\right) \log{\left (x \right )} - \frac{1}{x^{2}} \left(\log{\left (x \right )} - 2\right)^{3} \log^{3}{\left (x \right )}\right)$$