Производная x^e^atan(x)

Вы ввели

 / atan(x)\
 \E       /
x          

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

2. Теперь упростим:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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