Vamos aplicar:
y^' = ((1)/(36) x (9 x^2 - 2) sqrt(4 - 9 x^2) + (2)/(27) arcsen (3 x)/(2))^' = ((1)/(36) x (9 x^2 - 2) sqrt(4 - 9 x^2))^' + ((2)/(27) arcsen (3 x)/(2))^' = ((1)/(36) (9 x^3 - 2 x) sqrt(4 - 9 x^2))^' + ((2)/(27) arcsen (3 x)/(2))^'
Vamos resolver a primeira derivada:
((1)/(36) (9 x^3 - 2 x) sqrt(4 - 9 x^2))^' = f^' (x) g (x) + g^' (x) f (x) = ((1)/(36) (9 x^3 - 2 x))^' (sqrt(4 - 9 x^2)) + (sqrt(4 - 9 x^2))^' ((1)/(36) (9 x^3 - 2 x)) = ((1)/(36) (3 * 9 x^(3 - 1) - 2 x^(1 - 1))) (sqrt(4 - 9 x^2)) + ((1)/(2) (4 - 9 x^2)^((1)/(2) - 1) (4 - 9 x^2)^') ((1)/(36) (9 x^3 - 2 x)) = ((1)/(36) (27 x^2 - 2 x^0)) (sqrt(4 - 9 x^2)) + ((1)/(2) (4 - 9 x^2)^(- (1)/(2)) (0 - 2 * 9 x^(2 - 1))) ((1)/(36) (9 x^3 - 2 x)) = ((1)/(36) (27 x^2 - 2 * 1)) (sqrt(4 - 9 x^2)) + ((- 9 x)/(sqrt(4 - 9 x^2))) ((1)/(36) (9 x^3 - 2 x)) = ((27 x^2 - 2) sqrt(4 - 9 x^2))/(36) + (- 9 x (9 x^3 - 2 x))/(36 sqrt(4 - 9 x^2)) = ((27 x^2 - 2) (4 - 9 x^2) - 9 x (9 x^3 - 2 x))/(36 sqrt(4 - 9 x^2)) = (108 x^2 - 243 x^4 - 8 + 18 x^2 - 81 x^4 + 18 x^2)/(36 sqrt(4 - 9 x^2)) = (144 x^2 - 324 x^4 - 8)/(36 sqrt(4 - 9 x^2)) = (144 x^2 - 324 x^4 - 8)/(36 sqrt(4 - 9 x^2)) = (36 x^2 - 81 x^4 - 2)/(9 sqrt(4 - 9 x^2))
Agora a segunda:
((2)/(27) arcsen (3 x)/(2))^' = (2)/(27) * (1)/(sqrt(1 - ((3 x)/(2))^2)) * ((3 x)/(2))^' = (2)/(27 sqrt(1 - (9 x^2)/(4))) * ((3)/(2) x^(1 - 1)) = (2)/(27 sqrt((4 - 9 x^2)/(4))) * ((3)/(2) x^0) = (2)/((27 sqrt(4 - 9 x^2))/(sqrt(4))) * ((3)/(2) * 1) = (1)/((9 sqrt(4 - 9 x^2))/(2)) = (2)/(9 sqrt(4 - 9 x^2))
