Производная (acot(x))^asin(x)

Вы ввели

    asin(x)   
acot       (x)

$$\operatorname{acot}^{\operatorname{asin}{\left (x \right )}}{\left (x \right )}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

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Ответ:

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

Первая производная

    asin(x)    /log(acot(x))       asin(x)     \
acot       (x)*|------------ - ----------------|
               |   ________    /     2\        |
               |  /      2     \1 + x /*acot(x)|
               \\/  1 - x                      /

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

Вторая производная

               /                                   2                                                                                         \
    asin(x)    |/  log(acot(x))       asin(x)     \    x*log(acot(x))        asin(x)                      2                    2*x*asin(x)   |
acot       (x)*||- ------------ + ----------------|  + -------------- - ------------------ - ---------------------------- + -----------------|
               ||     ________    /     2\        |             3/2             2                        ________                   2        |
               ||    /      2     \1 + x /*acot(x)|     /     2\        /     2\      2      /     2\   /      2            /     2\         |
               \\  \/  1 - x                      /     \1 - x /        \1 + x / *acot (x)   \1 + x /*\/  1 - x  *acot(x)   \1 + x / *acot(x)/

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

Третья производная

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

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