It will depend on what the divorce and child custody agreement states. It sounds like you haven't already started that process, I recommend you consult with an attorney and do so asap. If you want to split the day care expenses, then it will need to be included in the agreement. Which is the same thing regarding any other expenses for the child. Everything must be spelled out in the agreement. Out of pocket medical expenses (make sure you specify re: vision, dental, medical, prescription, psychological) also you need to address education related expenses re: private school and college. Car expenses should be addressed for the time that the child starts to drive.
Just my dos pesos..... Good Luck