When you do not see any screws on the doorknob, press a flat screwdriver into the thin slot on its shank, and pull the knob to remove it. Use the screwdriver to pry the metal trim from lockset, and access the hidden screws holding it in place.
Removing the two formerly hidden screws releases the plate on one side of the door and the knob and the trim on the opposite side. Release the latch by removing the two screws that hold it in place on the edge of the door and sliding the mechanism forward, through the hole. When replacing the doorknob, avoid over-tightening the hidden screws so the new knob does not bind.