Производная (atan(2*x))^x

Вы ввели

    x     
atan (2*x)

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

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

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

Вторая производная

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

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

Третья производная

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

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