Производная acos(sin(x))^(2)

Вы ввели

    2        
acos (sin(x))

$$\operatorname{acos}^{2}{\left (\sin{\left (x \right )} \right )}$$

Первая производная

-2*acos(sin(x))*cos(x)
----------------------
      _____________   
     /        2       
   \/  1 - sin (x)    

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

Вторая производная

  /       2                                  2                       \
  |    cos (x)      acos(sin(x))*sin(x)   cos (x)*acos(sin(x))*sin(x)|
2*|- ------------ + ------------------- - ---------------------------|
  |          2           _____________                       3/2     |
  |  -1 + sin (x)       /        2              /       2   \        |
  \                   \/  1 - sin (x)           \1 - sin (x)/        /

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

Третья производная

  /                                     2                        2                        2                  2       2                \       
  |  acos(sin(x))       3*sin(x)     cos (x)*acos(sin(x))   3*sin (x)*acos(sin(x))   3*cos (x)*sin(x)   3*cos (x)*sin (x)*acos(sin(x))|       
2*|---------------- + ------------ - -------------------- + ---------------------- + ---------------- - ------------------------------|*cos(x)
  |   _____________           2                     3/2                     3/2                    2                        5/2       |       
  |  /        2       -1 + sin (x)     /       2   \           /       2   \         /        2   \            /       2   \          |       
  \\/  1 - sin (x)                     \1 - sin (x)/           \1 - sin (x)/         \-1 + sin (x)/            \1 - sin (x)/          /       

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