Производная (asin(x))^(5*x)

Вы ввели

    5*x   
asin   (x)

$$\operatorname{asin}^{5 x}{\left(x \right)}$$

d /    5*x   \
--\asin   (x)/
dx            

$$\frac{d}{d x} \operatorname{asin}^{5 x}{\left(x \right)}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right)$

---

Ответ:

$\left(5 x\right)^{5 x} \left(\log{\left(5 x \right)} + 1\right)$

Первая производная

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

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

Вторая производная

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

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

Третья производная

             /                                                  3                                                  2                                                                         /                    2                        \\
             |                                               3*x               3               4*x              3*x                   2*x               /         x                        \ |     2             x                x        ||
             |                                           ----------- + ----------------- + ----------- - ------------------ + --------------------   15*|------------------- + 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)||
      5*x    |   /         x                        \    \1 - x /                          \1 - x /      \-1 + x / *asin(x)   \1 - x /   *asin (x)      \\/  1 - x  *asin(x)               / \\/  1 - x     \1 - x /                       /|
5*asin   (x)*|25*|------------------- + log(asin(x))|  + ----------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------|
             |   |   ________                       |                                             asin(x)                                                                                    asin(x)                                        |
             |   |  /      2                        |                                                                                                                                                                                       |
             \   \\/  1 - x  *asin(x)               /                                                                                                                                                                                       /

$$5 \cdot \left(25 \left(\frac{x}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}} + \log{\left(\operatorname{asin}{\left(x \right)} \right)}\right)^{3} + \frac{15 \left(\frac{x}{\sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}} + \log{\left(\operatorname{asin}{\left(x \right)} \right)}\right) \left(\frac{x^{2}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{x}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}} + \frac{2}{\sqrt{1 - x^{2}}}\right)}{\operatorname{asin}{\left(x \right)}} + \frac{\frac{3 x^{3}}{\left(1 - x^{2}\right)^{\frac{5}{2}}} - \frac{3 x^{2}}{\left(x^{2} - 1\right)^{2} \operatorname{asin}{\left(x \right)}} + \frac{4 x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{2 x}{\left(1 - x^{2}\right)^{\frac{3}{2}} \operatorname{asin}^{2}{\left(x \right)}} + \frac{3}{\left(x^{2} - 1\right) \operatorname{asin}{\left(x \right)}}}{\operatorname{asin}{\left(x \right)}}\right) \operatorname{asin}^{5 x}{\left(x \right)}$$

График