I'm having a problem with the changing variable procedure for double integrals:

I was told to do that thing with the Jacobian etc. and it was proved to me that that was so because de linear transformation transforms areas and that the amount by how much the area is changed is the determinante of the matrix. Ok I can accept but I was wondering...can I prove that using differentials, I mean, changing dydx to dvdt I have that dy=(dy/dt)dt+(dy/dv)dv and dx=(dx/dt)dt+(dx/dv)dv so if I'm not wrong dydx=2(dydx/dtdt)dtdt+[(dy/dv)(dx/dt)+(dx/dv)(dy/dt)]dvdt=-[(dy/dv)(dx/dt)+(dx/dv)(dy/dt)]dvdt well that is not the jacobian I was talking about...what's wrong with that? where is my mistake?