Производная x^(atan(2*x))

Вы ввели

 atan(2*x)
x         

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

           /                      2                                         \
 atan(2*x) |/atan(2*x)   2*log(x)\    atan(2*x)        4         16*x*log(x)|
x         *||--------- + --------|  - --------- + ------------ - -----------|
           ||    x              2|         2        /       2\             2|
           |\            1 + 4*x /        x       x*\1 + 4*x /   /       2\ |
           \                                                     \1 + 4*x / /

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

Третья производная

           /                      3                                                                                                                                      2       \
 atan(2*x) |/atan(2*x)   2*log(x)\         48        16*log(x)          6           /atan(2*x)   2*log(x)\ /atan(2*x)        4         16*x*log(x)\   2*atan(2*x)   256*x *log(x)|
x         *||--------- + --------|  - ----------- - ----------- - ------------- - 3*|--------- + --------|*|--------- - ------------ + -----------| + ----------- + -------------|
           ||    x              2|              2             2    2 /       2\     |    x              2| |     2        /       2\             2|         3                  3 |
           |\            1 + 4*x /    /       2\    /       2\    x *\1 + 4*x /     \            1 + 4*x / |    x       x*\1 + 4*x /   /       2\ |        x         /       2\  |
           \                          \1 + 4*x /    \1 + 4*x /                                             \                           \1 + 4*x / /                  \1 + 4*x /  /

$$x^{\operatorname{atan}{\left (2 x \right )}} \left(\frac{256 x^{2} \log{\left (x \right )}}{\left(4 x^{2} + 1\right)^{3}} + \left(\frac{2 \log{\left (x \right )}}{4 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (2 x \right )}\right)^{3} - 3 \left(\frac{2 \log{\left (x \right )}}{4 x^{2} + 1} + \frac{1}{x} \operatorname{atan}{\left (2 x \right )}\right) \left(\frac{16 x \log{\left (x \right )}}{\left(4 x^{2} + 1\right)^{2}} - \frac{4}{x \left(4 x^{2} + 1\right)} + \frac{1}{x^{2}} \operatorname{atan}{\left (2 x \right )}\right) - \frac{16 \log{\left (x \right )}}{\left(4 x^{2} + 1\right)^{2}} - \frac{48}{\left(4 x^{2} + 1\right)^{2}} - \frac{6}{x^{2} \left(4 x^{2} + 1\right)} + \frac{2}{x^{3}} \operatorname{atan}{\left (2 x \right )}\right)$$