Buses Ireland?

Aircoach bus (coach) 705X takes about four hours from DUB to Derry, leaving at two-hour intervals on a random late June afternoon. It stops in Belfast and a few other places. Probably your best bet.

Translink (the National operator of Northern Ireland) also run Goldline route X3 to Derry- https://www.translink.co.uk/timetables?routeHashCode=d8503a6d175d2a6438c328b690990745b9eed0824d2f64d5d6f0b5cf2f7832bf&transportMode=Bus&route=T0&direction=H

I have taken buses all over Ireland -- at least a half dozen trips -- though not one from Dublin to Derry. Here are the bus timetables: https://www.buseireann.ie/

You have two options: You could go directly from Dublin to Derry via bus or air. (I'm not sure if there is a Dublin to Derry train.)

The other is to go from Dublin to Belfast and then Belfast to Derry. If this interests you, you could take the train from Dublin to Belfast; last time I was in Ireland, there was a train practically every hour from Dublin to Belfast: https://www.irishrail.ie/en-ie/rail-fares-and-tickets/fares-info/dublin-belfast. That way, you will be able to see the most populous city (Dublin), second most (Belfast) and sixth most (Derry) in Ireland.

From Belfast, you'd have to check if there is a train or just a bus from Belfast to Derry.

I've been to Dublin and Derry each. They're beautiful cities. I missed seeing Belfast because of time issues and regret it. Friends who have visited Belfast rave about how beautiful it is, too.

Let's be accurate here. The train from Dublin to Belfast is broadly every TWO hours. Last summer some peak extras were run with different rolling stock than the normal Enterprise trains.
From Belfast Lanyon Place (the terminus of Dublin trains) hourly trains run to Derry, operated by Translink.
There are also several bus routes, run by Translink, from Belfast to Derry- hourly or better express direct routes and other indirect routes which take quite a bit longer.
The express route takes no longer than the train.
The buses run from Europa Bus Station/Great Victoria Street railway station. From Dublin by train you would have to change at Lanyon Place to get there.
So in that case the Derry train is the more logical.
But you would have to go from Dublin Airport to Connolly Station to catch the train.
Three different companies run regular buses from Dublin Airport to Belfast Europa (or Glengall Street outside Europa).
There are NO Dublin to Derry flights.
Bus Eireann do NOT run Dublin to Derry buses, rather the two companies given above.

Note that Aircoach have now revised their routes and vehicle allocations. The 705X from DUB to Derry no longer has on board toilets. Brief 10 minute comfort stops (traffic permitting) are now made at Belfast International Airport and at Belfast Glengall Street (to use the facilities in the Europa Bus Station).

If you are going directly to Derry after arriving in Dublin, I'm going to suggest you consider the train.

Yes, you do have to go into Dublin to get it....but.....you've just been on a long flight across the ocean. Do you really want to sit on a coach, with just rest stops for bathroom breaks, and no opportunity to move around for four plus hours or would you prefer the train where you can get up, stretch your legs, and use the facilities anytime you want.

For completeness, as the question has unexpectedly arisen, on a tangent from another query about buses to Galway- the Translink X3 and X4 buses to Derry do not have on board toilets either- relying on a couple of rest stops.

On a long bus ride like this, forgetting other subjective matters like perceived comfort, that would make the difference to me to pivot me towards using the train.

You don't have to go into Dublin to catch the Belfast bound train. There is a frequent airport bus to Drogheda station, every 20 minutes, where you could pick the Belfast train up. There are lots of options.