Производная (asin(x))^(tan(x))

Вы ввели

    tan(x)   
asin      (x)

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

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

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

Вторая производная

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

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

Третья производная

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