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The court will order that you pay a specific amount based on your income and a few other relevant factors. You can calculate what you should be paying in support at the following link: http://www.childsupportillinois.com/assets/hfsweb016.pdf or http://www.alllaw.com/calculators/childsupport/illinois/
The best thing you can do is get a court order so there is no question about how much you should pay.
I understand that, but it has nothing to do with your child support obligation. The change definitely has caused you additional tax liability, but that's immaterial in this instance - legally.
All you can do is ask the court to allow you to claim the child in the future.