In normal clothes you ca certainly ride 10 miles each way. The trick is to ride slow enough so as you don't sweat. I used to do it 12 miles each way, carrying my jacket and work shoes in a pannier and wearing a breathable top. If it rains you do need a cape and lower leg gaiters, and the cape will slow you down a lot.
I usually rode in in cycling gear and showered and changed before work, it meant I could ride much faster and save most of the time it took to do this. I used to get a coffee and sit at my desk waiting for log-in time while colleagues cruised the works car park looking for a space.