Производная |x|^(1/|x|)

Вы ввели

    1 
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   |x|
|x|   

$$\left|{x}\right|^{\frac{1}{\left|{x}\right|}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

    1  /d                 d                 /d                 d              \         \
   --- |--(im(x))*im(x) + --(re(x))*re(x)   |--(im(x))*im(x) + --(re(x))*re(x)|*log(|x|)|
   |x| |dx                dx                \dx                dx             /         |
|x|   *|--------------------------------- - --------------------------------------------|
       |                  3                                        3                    |
       \               |x|                                      |x|                     /

$$\left(- \frac{1}{\left|{x}\right|^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \log{\left (\left|{x}\right| \right )} + \frac{1}{\left|{x}\right|^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)\right) \left|{x}\right|^{\frac{1}{\left|{x}\right|}}$$

Вторая производная

       /                                                                                                                                                                                        2                                                       2                                        2         \
    1  |                                                                                                                                                     /d                 d              \                   2 /d                 d              \      /d                 d              \          |
   --- |           2              2     2                  2                /           2              2     2                  2             \            4*|--(im(x))*im(x) + --(re(x))*re(x)|    (-1 + log(|x|)) *|--(im(x))*im(x) + --(re(x))*re(x)|    3*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)|
   |x| |/d        \    /d        \     d                  d                 |/d        \    /d        \     d                  d              |              \dx                dx             /                     \dx                dx             /      \dx                dx             /          |
|x|   *||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x) - ||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|) - -------------------------------------- + ----------------------------------------------------- + -----------------------------------------------|
       |\dx       /    \dx       /      2                  2                |\dx       /    \dx       /      2                  2             |                                2                                                3                                                    2                     |
       \                              dx                 dx                 \                              dx                 dx              /                             |x|                                              |x|                                                  |x|                      /
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                                                                                                                                                    |x|                                                                                                                                                     

$$\frac{\left|{x}\right|^{\frac{1}{\left|{x}\right|}}}{\left|{x}\right|^{3}} \left(\frac{1}{\left|{x}\right|^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \left(\log{\left (\left|{x}\right| \right )} - 1\right)^{2} + \frac{3}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \log{\left (\left|{x}\right| \right )} - \frac{4}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} - \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \log{\left (\left|{x}\right| \right )} + \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right)$$

Третья производная

        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        /                                                                                                                                                                                        2                                        2         \\ 
        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                                                                                                                     /d                 d              \      /d                 d              \          || 
        |                                                                                                                                                                                                                                                                                                                                    /           2              2     2                  2             \                                                                                            /           2              2     2                  2             \                                                                  |           2              2     2                  2                /           2              2     2                  2             \            4*|--(im(x))*im(x) + --(re(x))*re(x)|    3*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)|| 
        |                                                                                                                                                                               3                                                                                                         3      /d                 d              \ |/d        \    /d        \     d                  d              |                                         3              /d                 d              \ |/d        \    /d        \     d                  d              |                              /d                 d              \ |/d        \    /d        \     d                  d                 |/d        \    /d        \     d                  d              |              \dx                dx             /      \dx                dx             /          || 
     1  |                                                                                                                                            /d                 d              \                                                                     3 /d                 d              \    12*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|      /d                 d              \             9*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|)   3*(-1 + log(|x|))*|--(im(x))*im(x) + --(re(x))*re(x)|*||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x) - ||--(im(x))|  + |--(re(x))|  + ---(im(x))*im(x) + ---(re(x))*re(x)|*log(|x|) - -------------------------------------- + -----------------------------------------------|| 
    --- |/  3                  3                              2                        2       \              3                  3                23*|--(im(x))*im(x) + --(re(x))*re(x)|                  2                        2          (-1 + log(|x|)) *|--(im(x))*im(x) + --(re(x))*re(x)|       \dx                dx             / |\dx       /    \dx       /      2                  2             |   15*|--(im(x))*im(x) + --(re(x))*re(x)| *log(|x|)     \dx                dx             / |\dx       /    \dx       /      2                  2             |                              \dx                dx             / |\dx       /    \dx       /      2                  2                |\dx       /    \dx       /      2                  2             |                                2                                             2                     || 
    |x| || d                  d                   d          d             d          d        |             d                  d                    \dx                dx             /      d          d             d          d                            \dx                dx             /                                           \                              dx                 dx              /      \dx                dx             /                                                   \                              dx                 dx              /                                                                  \                              dx                 dx                 \                              dx                 dx              /                             |x|                                           |x|                      /| 
-|x|   *||---(im(x))*im(x) + ---(re(x))*re(x) + 3*--(im(x))*---(im(x)) + 3*--(re(x))*---(re(x))|*log(|x|) - ---(im(x))*im(x) - ---(re(x))*re(x) - --------------------------------------- - 3*--(im(x))*---(im(x)) - 3*--(re(x))*---(re(x)) + ----------------------------------------------------- + ---------------------------------------------------------------------------------------------------------- + ------------------------------------------------ - ------------------------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| 
        ||  3                  3                  dx          2            dx          2       |              3                  3                                     4                      dx          2            dx          2                                      6                                                                                 2                                                                               4                                                                                   2                                                                                                                                                                                                             3                                                                                                                                               | 
        \\dx                 dx                             dx                       dx        /            dx                 dx                                   |x|                                 dx                       dx                                    |x|                                                                               |x|                                                                             |x|                                                                                 |x|                                                                                                                                                                                                           |x|                                                                                                                                                / 
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$$- \frac{\left|{x}\right|^{\frac{1}{\left|{x}\right|}}}{\left|{x}\right|^{3}} \left(\frac{1}{\left|{x}\right|^{6}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} \left(\log{\left (\left|{x}\right| \right )} - 1\right)^{3} + \frac{15}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} \log{\left (\left|{x}\right| \right )} - \frac{23}{\left|{x}\right|^{4}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{3} + \frac{3}{\left|{x}\right|^{3}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\log{\left (\left|{x}\right| \right )} - 1\right) \left(\frac{3}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} \log{\left (\left|{x}\right| \right )} - \frac{4}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right)^{2} - \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \log{\left (\left|{x}\right| \right )} + \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) - \frac{9}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) \log{\left (\left|{x}\right| \right )} + \frac{12}{\left|{x}\right|^{2}} \left(\Re{x} \frac{d}{d x} \Re{x} + \Im{x} \frac{d}{d x} \Im{x}\right) \left(\Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} + \frac{d}{d x} \Re{x}^{2} + \frac{d}{d x} \Im{x}^{2}\right) + \left(\Re{x} \frac{d^{3}}{d x^{3}} \Re{x} + \Im{x} \frac{d^{3}}{d x^{3}} \Im{x} + 3 \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + 3 \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x}\right) \log{\left (\left|{x}\right| \right )} - \Re{x} \frac{d^{3}}{d x^{3}} \Re{x} - \Im{x} \frac{d^{3}}{d x^{3}} \Im{x} - 3 \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} - 3 \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x}\right)$$