Производная x^(atan(x))

Вы ввели

 atan(x)
x       

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

         /                  3                                                                                                                   2       \
 atan(x) |/atan(x)   log(x)\        6            3          /atan(x)   log(x)\ /atan(x)       2        2*x*log(x)\    2*log(x)   2*atan(x)   8*x *log(x)|
x       *||------- + ------|  - --------- - ----------- - 3*|------- + ------|*|------- - ---------- + ----------| - --------- + --------- + -----------|
         ||   x           2|            2    2 /     2\     |   x           2| |    2       /     2\           2 |           2        3               3 |
         |\          1 + x /    /     2\    x *\1 + x /     \          1 + x / |   x      x*\1 + x /   /     2\  |   /     2\        x        /     2\  |
         \                      \1 + x /                                       \                       \1 + x /  /   \1 + x /                 \1 + x /  /

$$x^{\operatorname{atan}{\left (x \right )}} \left(\frac{8 x^{2} \log{\left (x \right )}}{\left(x^{2} + 1\right)^{3}} + \left(\frac{\log{\left (x \right )}}{x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (x \right )}\right)^{3} - 3 \left(\frac{\log{\left (x \right )}}{x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (x \right )}\right) \left(\frac{2 x \log{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} - \frac{2}{x \left(x^{2} + 1\right)} + \frac{1}{x^{2}} \operatorname{atan}{\left (x \right )}\right) - \frac{2 \log{\left (x \right )}}{\left(x^{2} + 1\right)^{2}} - \frac{6}{\left(x^{2} + 1\right)^{2}} - \frac{3}{x^{2} \left(x^{2} + 1\right)} + \frac{2}{x^{3}} \operatorname{atan}{\left (x \right )}\right)$$