You need to figure out what the liquid is. When the exhaust is cold water vapor in the exhaust (a normal byproduct of combustion) can condense before it gets out and it will drip. That's normal and it should stop once the exhaust system is hot enough that the water vapor no longer condenses. I would think that 15 minutes of driving would be ample to heat the exhaust to the point where it would no longer condense – unless you live someplace quite cold.
The other possibilities aren't so good. If the liquid is coolant that means you've got a leak in the cooling system that is somehow finding its way into the exhaust. I suppose it could be another liquid as well, so it would be worth trying to catch some of those drips and identifying them.
I don't know this, but I would expect that a well designed exhaust system wouldn't accumulate too much condensate. But I suppose one possibility on short drives is that condensate is pooling in the exhaust someplace and, assuming that your driveway is really "up" perhaps it is draining accumulated water. You might try letting the car idle and watching to see if any drops come out (catch them on some clean white paper towel if you can).