First off, your first derivatives are incorrect. You didn't use the chain rule properly.

(I will do it your way, but note that: , which makes the derivative easier to think about.)

There are two ways to proceed from here. I would suspect that the standard thing to do would be two write dy/dx as a function of x, rather than leaving it a function of t. But it depends on what you are shooting for. I would do it that way simply because its a bit shorter to take the next derivative. So we have:

So

and go from there.

Otherwise I believe your method is correct, simply that your derivatives weren't correct.

-Dan