Use Implicit Differentiation:
-8y(dy/dx) = 8x−8y(dydx)=8x
dy/dx = (-x)/ydydx=−xy
(d^2y)/dx^2 = d/dx(dy/dx)d2ydx2=ddx(dydx)
(d^2y)/dx^2= (d((-x)/y))/dxd2ydx2=d(−xy)dx
(d^2y)/dx^2= {-y - -x(dy/dx)}/y^2d2ydx2=−y−−x(dydx)y2
(d^2y)/dx^2= {(-y^2)/y - -x((-x)/y)}/y^2d2ydx2=−y2y−−x(−xy)y2
(d^2y)/dx^2= -{y^2/y + -x((-x)/y)}/y^2d2ydx2=−y2y+−x(−xy)y2
(d^2y)/dx^2= -{y^2/y + x^2/y}/y^2d2ydx2=−y2y+x2yy2
(d^2y)/dx^2= -{y^2 + x^2}/y^3d2ydx2=−y2+x2y3
From the original equation, y^2 + x^2 = 1 y2+x2=1:
(d^2y)/dx^2= -1/y^3d2ydx2=−1y3