xy+y^2=4
Differentiating the given function w.r.t. x we find:
yd/dx(x) +xdy/dx+ 2ydy/dx = 0
i.e. y * 1 +xdy/dx+ 2ydy/dx = 0
i.e. y + (x+ 2y) dy/dx = 0
i.e. dy/dx = -y/(x+ 2y)
Differentiating again w.r.t. x we find:
d^2y/dx^2 = [(x+ 2y)dy/dx(-y) + (-y) d/dx(x+ 2y)]/(x+ 2y)^2
d^2y/dx^2 = [-(x+ 2y) + (-y) (1+ 2dy/dx)]/(x+ 2y)^2
d^2y/dx^2 = [-x- 2y -y (1+ 2dy/dx)]/(x+ 2y)^2
d^2y/dx^2 = [-x- 2y -y - 2ydy/dx)]/(x+ 2y)^2
d^2y/dx^2 = [-x- 3y - 2ydy/dx)]/(x+ 2y)^2