Производная (asin(cos(x)))^atan(x)

Вы ввели

    atan(x)        
asin       (cos(x))

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

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

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

---

Ответ:

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

Первая производная

    atan(x)         /log(asin(cos(x)))           atan(x)*sin(x)       \
asin       (cos(x))*|----------------- - -----------------------------|
                    |           2           _____________             |
                    |      1 + x           /        2                 |
                    \                    \/  1 - cos (x) *asin(cos(x))/

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

Вторая производная

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

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

Третья производная

                    /                                                       3                                                                                 /                                                                 2                                                                     2                     \      2                                                               3                                                                         3                                        2                           2       3                                2                                2                                                              3                                                              \
    atan(x)         |  /  log(asin(cos(x)))           atan(x)*sin(x)       \    2*log(asin(cos(x)))     /  log(asin(cos(x)))           atan(x)*sin(x)       \ |2*x*log(asin(cos(x)))           atan(x)*cos(x)                sin (x)*atan(x)                         2*sin(x)                      sin (x)*atan(x)*cos(x)   |   8*x *log(asin(cos(x)))           atan(x)*sin(x)                 sin (x)*atan(x)                         3*cos(x)                        2*sin (x)*atan(x)                        3*sin (x)                   3*cos (x)*sin (x)*atan(x)                3*sin (x)*cos(x)                 3*cos (x)*atan(x)*sin(x)       3*atan(x)*cos(x)*sin(x)         3*sin (x)*atan(x)*cos(x)                    6*x*sin(x)              |
asin       (cos(x))*|- |- ----------------- + -----------------------------|  - ------------------- + 3*|- ----------------- + -----------------------------|*|--------------------- + ----------------------------- - ---------------------------- + -------------------------------------- - -----------------------------| + ---------------------- + ----------------------------- - ----------------------------- - -------------------------------------- - ------------------------------ + ------------------------------------- - ----------------------------- + -------------------------------------- + ----------------------------- + ---------------------------- + ----------------------------- + ---------------------------------------|
                    |  |             2           _____________             |                 2          |             2           _____________             | |              2            _____________                /        2   \     2                       _____________                             3/2             |                 3             _____________                             3/2                            _____________                             3/2                 /     2\ /        2   \     2                        5/2                                      3/2                             3/2                /        2   \     2                         2                         2    _____________             |
                    |  |        1 + x           /        2                 |         /     2\           |        1 + x           /        2                 | |      /     2\            /        2                    \-1 + cos (x)/*asin (cos(x))   /     2\   /        2                    /       2   \                |         /     2\             /        2                    /       2   \                   /     2\   /        2                    /       2   \        3           \1 + x /*\-1 + cos (x)/*asin (cos(x))   /       2   \                   /     2\ /       2   \                   /       2   \                   \-1 + cos (x)/*asin (cos(x))   /        2   \      2           /     2\    /        2                 |
                    \  \                      \/  1 - cos (x) *asin(cos(x))/         \1 + x /           \                      \/  1 - cos (x) *asin(cos(x))/ \      \1 + x /          \/  1 - cos (x) *asin(cos(x))                                  \1 + x /*\/  1 - cos (x) *asin(cos(x))   \1 - cos (x)/   *asin(cos(x))/         \1 + x /           \/  1 - cos (x) *asin(cos(x))   \1 - cos (x)/   *asin(cos(x))   \1 + x /*\/  1 - cos (x) *asin(cos(x))   \1 - cos (x)/   *asin (cos(x))                                           \1 - cos (x)/   *asin(cos(x))   \1 + x /*\1 - cos (x)/   *asin(cos(x))   \1 - cos (x)/   *asin(cos(x))                                  \-1 + cos (x)/ *asin (cos(x))   \1 + x / *\/  1 - cos (x) *asin(cos(x))/

$$\left(\frac{8 x^{2}}{\left(x^{2} + 1\right)^{3}} \log{\left (\operatorname{asin}{\left (\cos{\left (x \right )} \right )} \right )} + \frac{6 x \sin{\left (x \right )}}{\left(x^{2} + 1\right)^{2} \sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \left(\frac{\sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{1}{x^{2} + 1} \log{\left (\operatorname{asin}{\left (\cos{\left (x \right )} \right )} \right )}\right)^{3} + 3 \left(\frac{\sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{1}{x^{2} + 1} \log{\left (\operatorname{asin}{\left (\cos{\left (x \right )} \right )} \right )}\right) \left(\frac{2 x}{\left(x^{2} + 1\right)^{2}} \log{\left (\operatorname{asin}{\left (\cos{\left (x \right )} \right )} \right )} - \frac{\sin^{2}{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} - 1\right) \operatorname{asin}^{2}{\left (\cos{\left (x \right )} \right )}} + \frac{\cos{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{\sin^{2}{\left (x \right )} \cos{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} + \frac{2 \sin{\left (x \right )}}{\left(x^{2} + 1\right) \sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}}\right) + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} - 1\right) \operatorname{asin}^{2}{\left (\cos{\left (x \right )} \right )}} + \frac{3 \sin^{3}{\left (x \right )} \cos{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} - 1\right)^{2} \operatorname{asin}^{2}{\left (\cos{\left (x \right )} \right )}} + \frac{\sin{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{\sin^{3}{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{2 \sin^{3}{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{3}{2}} \operatorname{asin}^{3}{\left (\cos{\left (x \right )} \right )}} + \frac{3 \sin{\left (x \right )} \cos^{2}{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{3 \sin^{3}{\left (x \right )} \cos^{2}{\left (x \right )} \operatorname{atan}{\left (x \right )}}{\left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{5}{2}} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} + \frac{3 \sin^{2}{\left (x \right )}}{\left(x^{2} + 1\right) \left(\cos^{2}{\left (x \right )} - 1\right) \operatorname{asin}^{2}{\left (\cos{\left (x \right )} \right )}} - \frac{3 \cos{\left (x \right )}}{\left(x^{2} + 1\right) \sqrt{- \cos^{2}{\left (x \right )} + 1} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} + \frac{3 \sin^{2}{\left (x \right )} \cos{\left (x \right )}}{\left(x^{2} + 1\right) \left(- \cos^{2}{\left (x \right )} + 1\right)^{\frac{3}{2}} \operatorname{asin}{\left (\cos{\left (x \right )} \right )}} - \frac{2}{\left(x^{2} + 1\right)^{2}} \log{\left (\operatorname{asin}{\left (\cos{\left (x \right )} \right )} \right )}\right) \operatorname{asin}^{\operatorname{atan}{\left (x \right )}}{\left (\cos{\left (x \right )} \right )}$$