Given............y.x^2.y^2 - 16x = 8
⇒x^2.y^3 -16x = 8
if f ' = dy/dx and f '' = d^2y/dx^2
then.....
2xy^3 + 3x^2.y^2.f ' - 16 = 0 .......................(1)
⇒f ' = {16 - 2x.y^3}/3x^2.y^2........................(2)
from eqn (1)
2y^3 + 6xy^2.f ' + 6xy^2.f ' + 6yx^2.f '' = 0
⇒f '' = -y^2/3x^2 - 2.f'
from eqn(2)
f '' = -y^2/3x^2 - 2{16 - 2xy^3}/3x^2.y^2
⇒ f '' = {4xy^3 - y^4 - 32}/3x^2.y^2