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
