Производная (atanh(2*x))^x

Вы ввели

     x     
atanh (2*x)

$$\operatorname{atanh}^{x}{\left (2 x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

     x      /         2*x                           \
atanh (2*x)*|--------------------- + log(atanh(2*x))|
            |/       2\                             |
            \\1 - 4*x /*atanh(2*x)                  /

$$\left(\frac{2 x}{\left(- 4 x^{2} + 1\right) \operatorname{atanh}{\left (2 x \right )}} + \log{\left (\operatorname{atanh}{\left (2 x \right )} \right )}\right) \operatorname{atanh}^{x}{\left (2 x \right )}$$

Вторая производная

            /                                                 /          2                           \\
            |                                                 |       4*x                x           ||
            |                                               4*|1 - --------- + ----------------------||
            |                                           2     |            2   /        2\           ||
     x      |/                            2*x          \      \    -1 + 4*x    \-1 + 4*x /*atanh(2*x)/|
atanh (2*x)*||-log(atanh(2*x)) + ----------------------|  - ------------------------------------------|
            ||                   /        2\           |              /        2\                     |
            \\                   \-1 + 4*x /*atanh(2*x)/              \-1 + 4*x /*atanh(2*x)          /

$$\left(\left(\frac{2 x}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} - \log{\left (\operatorname{atanh}{\left (2 x \right )} \right )}\right)^{2} - \frac{1}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} \left(- \frac{16 x^{2}}{4 x^{2} - 1} + \frac{4 x}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} + 4\right)\right) \operatorname{atanh}^{x}{\left (2 x \right )}$$

Третья производная

            /                                                   /                           3                 2                                   \                                                  /          2                           \\
            |                                                   |            3          64*x              24*x                       4*x          |      /                            2*x          \ |       4*x                x           ||
            |                                                 4*|-16*x + ---------- + --------- - ---------------------- + -----------------------|   12*|-log(atanh(2*x)) + ----------------------|*|1 - --------- + ----------------------||
            |                                             3     |        atanh(2*x)           2   /        2\              /        2\      2     |      |                   /        2\           | |            2   /        2\           ||
     x      |  /                            2*x          \      \                     -1 + 4*x    \-1 + 4*x /*atanh(2*x)   \-1 + 4*x /*atanh (2*x)/      \                   \-1 + 4*x /*atanh(2*x)/ \    -1 + 4*x    \-1 + 4*x /*atanh(2*x)/|
atanh (2*x)*|- |-log(atanh(2*x)) + ----------------------|  - ------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------|
            |  |                   /        2\           |                                              2                                                                              /        2\                                           |
            |  \                   \-1 + 4*x /*atanh(2*x)/                                   /        2\                                                                               \-1 + 4*x /*atanh(2*x)                                |
            \                                                                                \-1 + 4*x / *atanh(2*x)                                                                                                                         /

$$\left(- \left(\frac{2 x}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} - \log{\left (\operatorname{atanh}{\left (2 x \right )} \right )}\right)^{3} + \frac{12}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} \left(\frac{2 x}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} - \log{\left (\operatorname{atanh}{\left (2 x \right )} \right )}\right) \left(- \frac{4 x^{2}}{4 x^{2} - 1} + \frac{x}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} + 1\right) - \frac{1}{\left(4 x^{2} - 1\right)^{2} \operatorname{atanh}{\left (2 x \right )}} \left(\frac{256 x^{3}}{4 x^{2} - 1} - \frac{96 x^{2}}{\left(4 x^{2} - 1\right) \operatorname{atanh}{\left (2 x \right )}} - 64 x + \frac{16 x}{\left(4 x^{2} - 1\right) \operatorname{atanh}^{2}{\left (2 x \right )}} + \frac{12}{\operatorname{atanh}{\left (2 x \right )}}\right)\right) \operatorname{atanh}^{x}{\left (2 x \right )}$$