Производная (atan(x))^(log(x))

Вы ввели

    log(x)   
atan      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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