Производная (asin(x))^cos(x)

Вы ввели

    cos(x)   
asin      (x)

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

d /    cos(x)   \
--\asin      (x)/
dx               

$$\frac{d}{d x} \operatorname{asin}^{\cos{\left(x \right)}}{\left(x \right)}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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

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

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