How much does it cost to change the CV joint?

It is fairly complicated. Depending on what you mean by broke. But if it started clicking when you turned the wheel to one side or the other, then it probably just wore out. It can be removed and replaced fairly easy. But it takes "some" mechanical knowledge to do it. If not look around. Find a good backyard mechanic who can do it for you. If you have a warranty, then for sure take it to the dealer. But you know what those warrants are like. You end up paying more money than you planned. Always happens.