Vamos então aplicar:
y^' = (- sqrt((2)/(3)) arctg sqrt((2 (3 - x))/(3 (x - 2))))^' = - sqrt((2)/(3)) * (1)/(1 + (sqrt((2 (3 - x))/(3 (x - 2))))^2) * (sqrt((2 (3 - x))/(3 (x - 2))))^'
Vamos resolver a derivada
(sqrt((2 (3 - x))/(3 (x - 2))))^'
:
(sqrt((2 (3 - x))/(3 (x - 2))))^' = (((2 (3 - x))/(3 (x - 2)))^((1)/(2)))^' = (1)/(2) ((2 (3 - x))/(3 (x - 2)))^((1)/(2) - 1) ((2 (3 - x))/(3 (x - 2)))^' = (1)/(3) ((2 (3 - x))/(3 (x - 2)))^(- (1)/(2)) ((3 - x)/(x - 2))^'
Aqui vamos resolver
((3 - x)/(x - 2))^'
:
((3 - x)/(x - 2))^' = (f^' (x) g (x) - f (x) g ' (x))/([g (x)]^2) = ((3 - x)^' (x - 2) - (3 - x) (x - 2)^')/([x - 2]^2) = ((0 - x^(1 - 1)) (x - 2) - (3 - x) (x^(1 - 1) - 0))/((x - 2)^2) = ((- x^0) (x - 2) - (3 - x) (x^0))/((x - 2)^2) = ((- 1) (x - 2) - (3 - x) (1))/((x - 2)^2) = (- x + 2 - 3 + x)/((x - 2)^2) = (- 1)/((x - 2)^2)
Substituindo:
(sqrt((2 (3 - x))/(3 (x - 2))))^' = (1)/(3) ((2 (3 - x))/(3 (x - 2)))^(- (1)/(2)) ((3 - x)/(x - 2))^' = (1)/(3 sqrt((2 (3 - x))/(3 (x - 2)))) ((- 1)/((x - 2)^2)) = (- 1)/(3 (x - 2)^2 sqrt((2 (3 - x))/(3 (x - 2)))) = (- 1)/(3 (x - 2)^2 (sqrt(2 (3 - x)))/(sqrt(3 (x - 2)))) = (- 1)/(3 (x - 2)^2 (sqrt(2) sqrt(3 - x))/(sqrt(3) sqrt(x - 2 ))) = (- 1)/((3 sqrt(2) sqrt(3 - x) (x - 2)^2)/(3^((1)/(2)) (x - 2)^((1)/(2)))) = (- 1)/(3^(1 - (1)/(2)) sqrt(2) sqrt(3 - x) (x - 2)^(2 - (1)/(2))) = (- 1)/(3^((1)/(2)) sqrt(2) sqrt(3 - x) (x - 2)^((3)/(2))) = (- 1)/(sqrt(3) sqrt(2) sqrt(3 - x) (x - 2)^((3)/(2))) = (- 1)/(sqrt(6) sqrt(3 - x) (x - 2)^((3)/(2)))