Q: How to track a child's location through their iphone

You would need to have find my iPhone on your phone and their phone on the same iCloud account. It would be the same as tracking a lost device.see this article for more info: iCloud: Locate your device You may also want to check out family sharing Family Sharing and location - Apple Support -AJ

There is an app called Find My Friends.  The app must be on both phones or iPad.  If you are attempting to track your child without their knowledge, adding FMF to their phone will be a challenge because an invitation has to be sent from you to them and they have to accept it.  Once installed on both phones, you are able to track your child's location as long as the phone is on.  Your child would also be able to track your location as well.