Производная (log(x))^atan(x)

Вы ввели

   atan(x)   
log       (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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