Find y'' by implicit differentiation. 4x3 + 5y3 = 2

Differentiate implicitly using the power rule:

12x^2 + 15y^2y' = 0

Make y' the subject:

15y^2y' = - 12x^2

y' = - (4x^2) / 3y^2

Differentiate again using the quotient rule:

y'' = [-8x (3y^2) - 6yy' (-4x^2) ] / 9y^4

y'' = (-24xy^2 + 24x^2yy') / 9y^4

Simplify:

y'' = (-8xy + 8x^2y') / 3y^3

Substitute (4x^2) / 3y^2 back for y':

y'' = (-8xy + 8x^2 * 4x^2 / 3y^2) / 3y^3

y'' = (-24xy^3 + 32x^4) / 9y^5