Enter rental address when registering your business?

I currently rent an apartment and wanted to earn a little extra with graphic design.
For this I would like to register a small business beforehand, but I need an address as a "permanent establishment". I would only work on the laptop / tablet, so I wouldn't bother neighbors with it and not even necessarily work at home. Would that be possible or do I have to ask my landlord? I thank you in advance.

No, in that case you don't need to ask anyone.

You only have to ask questions / register with the landlord if you now regularly receive customers, suppliers, visitors, clients, …, so you use the apartment / stairwell beyond the usual extent.

Or make noise and dirt.

In this case you are sitting at the PC, everything is normal, everything is fine.

Otherwise all PC Fuzzis would have to register their home office job with the landlord. Nobody does, nobody needs!

If you do your commercial activity from home, then your apartment is your business premises. And you don't need to ask the landlord for this activity.

By the way, you can only register one trade. There's no small business in trade law…

https://www.mietrecht.org/mietvertrag/gewerbe-mietwohnung/

b) These activities must be approved by the landlord
…

If the tenant uses his home address as his business address and his professional activity is thus exposed to the outside world, the landlord may not refuse to grant the necessary approval according to a judgment of the AG Hamburg - Blankenese dated November 20, 2012 - 532 C 259/12 - if the Specifying the residential address as the business address does not lead to the apartment being worn down more than by mere occupation and also to no nuisance to residents or neighbors because neither employees nor customers regularly enter the property, nor post deliveries or deliveries of goods lead to a greater impairment to lead.