I don't really follow your work, but...
where y is a function of x.
Can you solve for y'?
Need help to finish this question,also how do I post in TeXnicCenter?
Find dy/dx and d^2y/dx^2 for x^2y - sin(x^2 - y) = 20
2x. dy/dx - cos(x^2 - y) (2x - dy/dx) = 0
2x. dy/dx = cos(x^2 - y) (2x - dy/dx)
dy/dx = cos(x^2 - y) (2x - dy/dx)/2x
dy/dx . (dx/2x - dy) = cos(x^2 - y)/2x
can't factor out dy/dx
There seems to be a common problem in your implicit differentiation. For example, let's take your first term:
There are two terms here, so you need to use the product rule, not just take the derivative of both. ie:
On to your problem:
Taking the first derivative of both sides:
To get y'' it would probably be easier to pick one of the line above before the division. (To get around that pesky quotient rule.)
Take the next derivative:
(I have put  around derivatives of individual terms above.)
You can solve for y'' from here. It will get a bit messy at some point because you should replace the y' in your answer for y'' with the expression you first got above. (In other words sub in your solution for y' in your final answer so that you have y'' as a function of just x and y.)