Imagine you are a resident of California and have a California driver's license. If you decide to take a road trip to Arizona, which clause of the Constitution requires Arizona to recognize your California driver's license?

The answer to the question is Full faith and credit clause

This clause of the constitution requires that all the states within the United States should respect the "public acts, records, and judicial proceedings of every other state.", and for this reason Arizona would have to recognize a drivers licence issued by the state of California.