(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:
and go from there.
Otherwise I believe your method is correct, simply that your derivatives weren't correct.