When saying 'hjemme' it's just general and I'd translate it to 'at home'. Vi er hjemme nu - We are at home now When you use 'derhjemme' it's when you're talking about your home, but you are not at home. 'derhjemme's literal translation would something 'there at home': 'back home' is the more correct translation i think. So if you're meeting with a friend at a cafe and you just got a new dog you'd say Vi har fået en ny hund derhjemme - We got a new dog at home There is also a third one 'herhjemme' which is the "opposite" of 'derhjemme' and it means 'in this house' or more literally 'here at home' Herhjemme står vi op klokken 06.00 - In this house we wake up at 06.00 One last thing to add is that 'derhjemme' can also be used in the same way as in english with 'back home' in regards to country or region. Her i Sverige spiser de köttbullar, men derhjemme spiser vi frikadeller - Here in Sweden they eat köttbullar, but back home we eat frikadeller. https://ordnet.dk/ddo/ordbog?query=derhjemme (the danish dictionary's explanation, cause I had to look it up because I have never thought of it. Also the description is a bit weird even for a dane there)

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