Производная tan(x)^asin(x)

Вы ввели

   asin(x)   
tan       (x)

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

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

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

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

Вторая производная

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

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

Третья производная

              /                                     3                                                         /                                                       2                             \                                    2                            2                   3                                                                                 \
              |/              /       2   \        \                    /              /       2   \        \ |                                          /       2   \               /       2   \  |     /       2   \     /       2   \                /       2   \       /       2   \               2                                                    /       2   \ |
   asin(x)    ||log(tan(x))   \1 + tan (x)/*asin(x)|    log(tan(x))     |log(tan(x))   \1 + tan (x)/*asin(x)| |  /       2   \           x*log(tan(x))   \1 + tan (x)/ *asin(x)    2*\1 + tan (x)/  |   6*\1 + tan (x)/   4*\1 + tan (x)/ *asin(x)     3*\1 + tan (x)/     2*\1 + tan (x)/ *asin(x)   3*x *log(tan(x))     /       2   \                  3*x*\1 + tan (x)/ |
tan       (x)*||----------- + ---------------------|  + ----------- + 3*|----------- + ---------------------|*|2*\1 + tan (x)/*asin(x) + ------------- - ---------------------- + ------------------| + --------------- - ------------------------ - ------------------- + ------------------------ + ---------------- + 4*\1 + tan (x)/*asin(x)*tan(x) + ------------------|
              ||   ________           tan(x)       |            3/2     |   ________           tan(x)       | |                                   3/2              2                 ________       |        ________              tan(x)               ________                      3                         5/2                                               3/2       |
              ||  /      2                         |    /     2\        |  /      2                         | |                           /     2\              tan (x)             /      2        |       /      2                                   /      2     2              tan (x)              /     2\                                          /     2\          |
              \\\/  1 - x                          /    \1 - x /        \\/  1 - x                          / \                           \1 - x /                                \/  1 - x  *tan(x)/     \/  1 - x                                  \/  1 - x  *tan (x)                                \1 - x /                                          \1 - x /   *tan(x)/

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