Second derivatives using implicit differentiation

Hi there. Well, I wanted to know how to find the second derivatives of a function using implicit differentiation. Is it possible? I think it is. I think I must use the chain rule somehow, but I don't know how... I'm in multivariable calculus since the function I'm gonna use could be seen as a function of only one variable.

An ellipse:

So we have the partial derivatives:

,

,

So then, using implicit differentiation:

But now if I wanna find how should I proceed?