So I emailed a professor asking the professor about taking a test earlier than the scheduled day it is on next week and the professor responded and said they could arrange for me to take the test early if I want. And to let the professor know what I have in mind.

My question is: Should I tell the professor in the response why I want to take the test early? The reason I want to take it early is because I have a doctor appointment during class time the day of the test. If, I should tell the professor why the reason I need to take the test early, should I also tell the professor what the doctor appointment is for or no?

No, you need not reveal private information. I faced a similar situation once. If the professor had denied you the right to take the exam early, you could argue about the doctor's appointment. But since you have permission to take the exam early, let things lie as they are. What you have in mind is arrangements for taking the test (time and place, etc.). That seems to be what he or she is asking for.

Whether or not you should tell the professor why you want to take the test early is ultimately your decision. However, providing a valid and legitimate reason can help your case and increase the chances of the professor accommodating your request. In this situation, it is appropriate to share that you have a doctor's appointment during the class time on the day of the test. It is not necessary, however, to disclose the specific details of the appointment unless you are comfortable doing so.

When responding to the professor, you can mention that you have a conflicting doctor's appointment on the day of the test and express your gratitude for their willingness to arrange an alternative time. You may also consider providing any necessary documentation, such as a doctor's note, to support your request.

To summarize, it is generally recommended to explain your valid reason for requesting to take the test early, but disclosing specific details is optional and entirely up to your discretion.