Vamos aplicar:
y^' = ((x^3)/(3) arcsen x + (x^2 + 2)/(9) sqrt(1 - x^2))^' = ((x^3)/(3) arcsen x)^' + ((x^2 + 2)/(9) sqrt(1 - x^2))^'
Vamos resolver a primeira derivada:
((x^3)/(3) arcsen x)^' = f^' (x) g (x) + g^' (x) f (x) = ((x^3)/(3))^' (arcsen x) + (arcsen x)^' ((x^3)/(3)) = (3 x^(3 - 1))/(3) (arcsen x) + (1)/(sqrt(1 - x^2)) ((x^3)/(3)) = x^2 arcsen x + (x^3)/(3 sqrt(1 - x^2))
Agora a segunda:
((x^2 + 2)/(9) sqrt(1 - x^2))^' = f^' (x) g (x) + g^' (x) f (x) = ((x^2 + 2)/(9))^' (sqrt(1 - x^2)) + (sqrt(1 - x^2))^' ((x^2 + 2)/(9)) = ((2 x^(2 - 1) + 0)/(9)) (sqrt(1 - x^2)) + ((1)/(2) (1 - x^2)^((1)/(2) - 1) (1 - x^2)^') ((x^2 + 2)/(9)) = ((2 x)/(9)) (sqrt(1 - x^2)) + ((1)/(2) (1 - x^2)^(- (1)/(2)) (0 - 2 x^(2 - 1))) ((x^2 + 2)/(9)) = (2 x sqrt(1 - x^2))/(9) + ((1)/(2 sqrt(1 - x^2)) (- 2 x)) ((x^2 + 2)/(9)) = (2 x sqrt(1 - x^2))/(9) - (x (x^2 + 2))/(9 sqrt(1 - x^2)) = (2 x sqrt(1 - x^2) sqrt(1 - x^2) - x (x^2 + 2))/(9 sqrt(1 - x^2)) = (2 x (1 - x^2) - x^3 - 2 x)/(9 sqrt(1 - x^2)) = (2 x - 2 x^3 - x^3 - 2 x)/(9 sqrt(1 - x^2)) = - (3 x^3)/(9 sqrt(1 - x^2)) = - (x^3)/(3 sqrt(1 - x^2))