Производная (acos(x))^(atan(x))

Вы ввели

    atan(x)   
acos       (x)

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

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

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

Вторая производная

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

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

Третья производная

               /                                    3                                                                                                                                                                                                                                          2                                                                            2                                           \
    atan(x)    |/log(acos(x))         atan(x)      \      /log(acos(x))         atan(x)      \ /       atan(x)         2*x*log(acos(x))                2                      x*atan(x)     \   2*log(acos(x))         atan(x)              2*atan(x)                      3                8*x *log(acos(x))               3*x                    3*x*atan(x)           3*x *atan(x)                   6*x             |
acos       (x)*||------------ - -------------------|  - 3*|------------ - -------------------|*|- ------------------ + ---------------- + ---------------------------- + -------------------| - -------------- - ------------------- - -------------------- + --------------------------- + ----------------- - ---------------------------- - ------------------- - ------------------- + -----------------------------|
               ||        2         ________        |      |        2         ________        | |  /      2\     2                 2                   ________                   3/2        |             2              3/2                   3/2            /     2\ /      2\     2                  3                        3/2                    2                    5/2                   2    ________        |
               ||   1 + x         /      2         |      |   1 + x         /      2         | |  \-1 + x /*acos (x)      /     2\        /     2\   /      2            /     2\           |     /     2\       /     2\              /     2\        3      \1 + x /*\-1 + x /*acos (x)       /     2\        /     2\ /     2\              /      2\      2      /     2\              /     2\    /      2         |
               \\               \/  1 - x  *acos(x)/      \               \/  1 - x  *acos(x)/ \                          \1 + x /        \1 + x /*\/  1 - x  *acos(x)   \1 - x /   *acos(x)/     \1 + x /       \1 - x /   *acos(x)   \1 - x /   *acos (x)                                     \1 + x /        \1 + x /*\1 - x /   *acos(x)   \-1 + x / *acos (x)   \1 - x /   *acos(x)   \1 + x / *\/  1 - x  *acos(x)/

$$\left(\frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} \log{\left (\operatorname{acos}{\left (x \right )} \right )} - \frac{3 x^{2} \operatorname{atan}{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{5}{2}} \operatorname{acos}{\left (x \right )}} - \frac{3 x \operatorname{atan}{\left (x \right )}}{\left(x^{2} - 1\right)^{2} \operatorname{acos}^{2}{\left (x \right )}} + \frac{6 x}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right)^{2} \operatorname{acos}{\left (x \right )}} - \frac{3 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \left(x^{2} + 1\right) \operatorname{acos}{\left (x \right )}} + \left(\frac{1}{x^{2} + 1} \log{\left (\operatorname{acos}{\left (x \right )} \right )} - \frac{\operatorname{atan}{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}\right)^{3} - 3 \left(\frac{1}{x^{2} + 1} \log{\left (\operatorname{acos}{\left (x \right )} \right )} - \frac{\operatorname{atan}{\left (x \right )}}{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}\right) \left(\frac{2 x}{\left(x^{2} + 1\right)^{2}} \log{\left (\operatorname{acos}{\left (x \right )} \right )} + \frac{x \operatorname{atan}{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} - \frac{\operatorname{atan}{\left (x \right )}}{\left(x^{2} - 1\right) \operatorname{acos}^{2}{\left (x \right )}} + \frac{2}{\sqrt{- x^{2} + 1} \left(x^{2} + 1\right) \operatorname{acos}{\left (x \right )}}\right) - \frac{2}{\left(x^{2} + 1\right)^{2}} \log{\left (\operatorname{acos}{\left (x \right )} \right )} + \frac{3}{\left(x^{2} - 1\right) \left(x^{2} + 1\right) \operatorname{acos}^{2}{\left (x \right )}} - \frac{\operatorname{atan}{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}{\left (x \right )}} - \frac{2 \operatorname{atan}{\left (x \right )}}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{acos}^{3}{\left (x \right )}}\right) \operatorname{acos}^{\operatorname{atan}{\left (x \right )}}{\left (x \right )}$$