Производная x^cot(1/x)

Вы ввели

    /1\
 cot|-|
    \x/
x      

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

        /                               3                                                                                                                                                                         2                                                                                                                           \
    /1\ |//       2/1\\                \                 /       2/1\\     //       2/1\\                \ /    /       2/1\\     /       2/1\\            /       2/1\\    /1\                \     /       2/1\\             /       2/1\\            /       2/1\\    /1\      /       2/1\\    /1\               2/1\ /       2/1\\       |
 cot|-| |||1 + cot |-||*log(x)         |               9*|1 + cot |-||     ||1 + cot |-||*log(x)         | |  2*|1 + cot |-||   2*|1 + cot |-||*log(x)   2*|1 + cot |-||*cot|-|*log(x)         |   2*|1 + cot |-|| *log(x)   6*|1 + cot |-||*log(x)   6*|1 + cot |-||*cot|-|   12*|1 + cot |-||*cot|-|*log(x)   4*cot |-|*|1 + cot |-||*log(x)|
    \x/ ||\        \x//             /1\|         /1\     \        \x//     |\        \x//             /1\| |    \        \x//     \        \x//            \        \x//    \x/             /1\|     \        \x//             \        \x//            \        \x//    \x/      \        \x//    \x/                \x/ \        \x//       |
x      *||-------------------- + cot|-||  + 2*cot|-| - --------------- - 3*|-------------------- + cot|-||*|- --------------- + ---------------------- - ----------------------------- + cot|-|| + ----------------------- + ---------------------- + ---------------------- - ------------------------------ + ------------------------------|
        |\         x                \x//         \x/          x            \         x                \x// |         x                    x                             2                   \x/|               3                       x                         2                            2                                3              |
        \                                                                                                  \                                                           x                       /              x                                                 x                            x                                x               /
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                                                                                                                                                                       x                                                                                                                                                                       

$$\frac{1}{x^{3}} x^{\cot{\left (\frac{1}{x} \right )}} \left(\left(\cot{\left (\frac{1}{x} \right )} + \frac{1}{x} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )}\right)^{3} - 3 \left(\cot{\left (\frac{1}{x} \right )} + \frac{1}{x} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )}\right) \left(\cot{\left (\frac{1}{x} \right )} + \frac{2}{x} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \left(2 \cot^{2}{\left (\frac{1}{x} \right )} + 2\right) - \frac{2}{x^{2}} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )} \cot{\left (\frac{1}{x} \right )}\right) + 2 \cot{\left (\frac{1}{x} \right )} + \frac{6}{x} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )} - \frac{1}{x} \left(9 \cot^{2}{\left (\frac{1}{x} \right )} + 9\right) - \frac{12}{x^{2}} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )} \cot{\left (\frac{1}{x} \right )} + \frac{6}{x^{2}} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \cot{\left (\frac{1}{x} \right )} + \frac{2}{x^{3}} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right)^{2} \log{\left (x \right )} + \frac{4}{x^{3}} \left(\cot^{2}{\left (\frac{1}{x} \right )} + 1\right) \log{\left (x \right )} \cot^{2}{\left (\frac{1}{x} \right )}\right)$$