d2x/dy2 equals

Let x = f(y).
Then dx/dy = f'(y) and d2x/dy2 = f''(y).
Also, let y = g(x).
Then dy/dx = g'(x) and d2y/dx2 = g''(x).
We have dy/dx = 1/(dx/dy).
Differentiating with respect to x, we get
d2y/dx2 = d/dx(1/(dx/dy)) = -1/(dx/dy)2 * d/dx(dx/dy)
= -1/(dx/dy)2 * d/dx(dx/dy)
= -1/(dx/dy)2 * d(dx/dy)/dy * dy/dx
= -1/(dx/dy)2 * (d2x/dy2) * (dy/dx)
= -(d2x/dy2)/(dx/dy)3
d2y/dx2 = - (d2x/dy2)(dy/dx)^3
Therefore, d2x/dy2 = -(d2y/dx2)(dx/dy)^3