Question: numerical solution to a non-linear system of equations including derivatives

Hello,

is there any way to solve a system like (d denotes the derivates):

f₁(x₁,x₂,y₁,y₂)=0

f₂(x₁,x₂,y₁,y₂)=0

dg₁(x₁,x₂,y₁,y₂)/dy₁=0

dg₂(x₁,x₂,y₁,y₂)/dy₂=0

If it were possible to solve the first two equations analytically, this would give me x1 and x2 depending on y1 and y2. Then I would insert this into the last two equations and solve them for y1 and y2. However, all equations are non-linear (and there are more than mentioned here) so that I don't get an analytical solution. The problem with numerical calculations is that I can't calculate the derivatives without having solved the first two equations analytically, i.e. I can't transform it into an ordinary system of four nonlinear equations. Is there any possibility to solve the system numerically anyway?

I would be very very grateful for any suggestions, since I have been struggling with this for several weeks now.

Thanks a lot,

Ciccarese