Производная cos(x)^asin(x)

Вы ввели

   asin(x)   
cos       (x)

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

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

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

Вторая производная

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

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

Третья производная

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

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

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