I worked the problem out twice and got the same answer both times. I think that it is right.
1) The second answer you wrote is correct.
2) There is a negative sign missing in your first solution when you took the derivative of csc(x).
3) Some of your notation leaves something to be desired. In both solutions you wrote:
d/dx[3x^2] * d/dx[-4x^2] * y
to mean the result:
6x - 8xy
The actual result is:
d/dx[3x^2] * d/dx[-4x^2] * y = (6x)*(8x)*y = 48(x^2)y
You really need to write:
d/dx[3x^2] - d/dx[4x^2] * y