Производная (asin(x)^x)^2

Вы ввели

        2
    x    
asin (x) 

$$\left(\operatorname{asin}^{x}{\left (x \right )}\right)^{2}$$

Подробное решение

1. Заменим $u = \operatorname{asin}^{x}{\left (x \right )}$.

2. В силу правила, применим: $u^{2}$ получим $2 u$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \operatorname{asin}^{x}{\left (x \right )}$:

   1. Не могу найти шаги в поиске этой производной.

      Но производная

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

   В результате последовательности правил:

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

             /                                                 3                                                  2                                                                        /                    2                        \\
             |                                              3*x               3               4*x              3*x                   2*x              /         x                        \ |     2             x                x        ||
             |                                          ----------- + ----------------- + ----------- - ------------------ + --------------------   6*|------------------- + log(asin(x))|*|----------- + ----------- + -----------------||
             |                                                  5/2   /      2\                   3/2            2                   3/2              |   ________                       | |   ________           3/2   /      2\        ||
             |                                      3   /     2\      \-1 + x /*asin(x)   /     2\      /      2\            /     2\        2        |  /      2                        | |  /      2    /     2\      \-1 + x /*asin(x)||
      2*x    |  /         x                        \    \1 - x /                          \1 - x /      \-1 + x / *asin(x)   \1 - x /   *asin (x)     \\/  1 - x  *asin(x)               / \\/  1 - x     \1 - x /                       /|
2*asin   (x)*|4*|------------------- + log(asin(x))|  + ----------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------|
             |  |   ________                       |                                             asin(x)                                                                                   asin(x)                                        |
             |  |  /      2                        |                                                                                                                                                                                      |
             \  \\/  1 - x  *asin(x)               /                                                                                                                                                                                      /

$$2 \left(4 \left(\frac{x}{\sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}} + \log{\left (\operatorname{asin}{\left (x \right )} \right )}\right)^{3} + \frac{6}{\operatorname{asin}{\left (x \right )}} \left(\frac{x}{\sqrt{- x^{2} + 1} \operatorname{asin}{\left (x \right )}} + \log{\left (\operatorname{asin}{\left (x \right )} \right )}\right) \left(\frac{x^{2}}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{x}{\left(x^{2} - 1\right) \operatorname{asin}{\left (x \right )}} + \frac{2}{\sqrt{- x^{2} + 1}}\right) + \frac{1}{\operatorname{asin}{\left (x \right )}} \left(\frac{3 x^{3}}{\left(- x^{2} + 1\right)^{\frac{5}{2}}} - \frac{3 x^{2}}{\left(x^{2} - 1\right)^{2} \operatorname{asin}{\left (x \right )}} + \frac{4 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{2 x}{\left(- x^{2} + 1\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left (x \right )}} + \frac{3}{\left(x^{2} - 1\right) \operatorname{asin}{\left (x \right )}}\right)\right) \operatorname{asin}^{2 x}{\left (x \right )}$$