Производная (atan(x))^tan(x)

Вы ввели

    tan(x)   
atan      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

              /                                               3                                                     /                       /       2   \                                                           \                  2                   /       2   \                                                                                         /       2   \                          /       2   \                2          \
    tan(x)    |//       2   \                     tan(x)     \      //       2   \                     tan(x)     \ |      tan(x)         2*\1 + tan (x)/      /       2   \                           2*x*tan(x)   |     /       2   \                  3*\1 + tan (x)/          2*tan(x)            2*tan(x)             2    /       2   \                6*x*\1 + tan (x)/       6*x*tan(x)       6*\1 + tan (x)/*tan(x)      8*x *tan(x)   |
atan      (x)*||\1 + tan (x)/*log(atan(x)) + ----------------|  - 3*|\1 + tan (x)/*log(atan(x)) + ----------------|*|------------------ - ---------------- - 2*\1 + tan (x)/*log(atan(x))*tan(x) + -----------------| + 2*\1 + tan (x)/ *log(atan(x)) - ------------------ - ----------------- + ------------------ + 4*tan (x)*\1 + tan (x)/*log(atan(x)) - ----------------- + ------------------ + ---------------------- + -----------------|
              ||                             /     2\        |      |                             /     2\        | |        2            /     2\                                                         2        |                                           2                    2                   3                                                           2                   3               /     2\                      3        |
              |\                             \1 + x /*atan(x)/      \                             \1 + x /*atan(x)/ |/     2\      2      \1 + x /*atan(x)                                         /     2\         |                                   /     2\      2      /     2\            /     2\      3                                             /     2\            /     2\      2         \1 + x /*atan(x)      /     2\         |
              \                                                                                                     \\1 + x / *atan (x)                                                            \1 + x / *atan(x)/                                   \1 + x / *atan (x)   \1 + x / *atan(x)   \1 + x / *atan (x)                                          \1 + x / *atan(x)   \1 + x / *atan (x)                            \1 + x / *atan(x)/

$$\left(\frac{8 x^{2} \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}{\left (x \right )}} - \frac{6 x \left(\tan^{2}{\left (x \right )} + 1\right)}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{6 x \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{2}{\left (x \right )}} + \left(\left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\operatorname{atan}{\left (x \right )} \right )} + \frac{\tan{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}}\right)^{3} - 3 \left(\left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\operatorname{atan}{\left (x \right )} \right )} + \frac{\tan{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}}\right) \left(\frac{2 x \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} - 2 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\operatorname{atan}{\left (x \right )} \right )} \tan{\left (x \right )} - \frac{2 \tan^{2}{\left (x \right )} + 2}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} + \frac{\tan{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}}\right) + 2 \left(\tan^{2}{\left (x \right )} + 1\right)^{2} \log{\left (\operatorname{atan}{\left (x \right )} \right )} + 4 \left(\tan^{2}{\left (x \right )} + 1\right) \log{\left (\operatorname{atan}{\left (x \right )} \right )} \tan^{2}{\left (x \right )} + \frac{6 \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )}}{\left(x^{2} + 1\right) \operatorname{atan}{\left (x \right )}} - \frac{3 \tan^{2}{\left (x \right )} + 3}{\left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left (x \right )}} - \frac{2 \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left (x \right )}} + \frac{2 \tan{\left (x \right )}}{\left(x^{2} + 1\right)^{3} \operatorname{atan}^{3}{\left (x \right )}}\right) \operatorname{atan}^{\tan{\left (x \right )}}{\left (x \right )}$$