Производная atan(x)^x

Вы ввели

    x   
atan (x)

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

d /    x   \
--\atan (x)/
dx          

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

График