Производная sin(x)^asin(x)

Вы ввели

   asin(x)   
sin       (x)

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

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

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

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

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

Вторая производная

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

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

Третья производная

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

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

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