To unclog a bathtub drain the right way, you'll need to take the drain plate off. That's the little round plate that has the lever on it for opening and closing the pop-up drain stopper. It takes a little effort, but it's not too hard to remove and behind that plate, you have access to a straight drain pipe you can send the drain snake down. Check out the video above from This Old House for instructions.
When a person has a blockage in their sewage system they often try to fix it themselves by adding an acid or a base such as Drano in an attempt to dissolve or dislodge the problem. These chemicals can get into the plumbers eyes when the sewage is splashed during the repair. The plumbers skin during the repair does come into contact with the sewage water. The owner of the toilet might not report to the plumber they have already tried Drano a highly caustic base .