Производная (atan(x))^(1/x)

Вы ввели

x _________
\/ atan(x) 

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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