A simple vinegar solution will clean, disinfect and deodorize your acrylic bathtub without causing damage to the finish. In a spray bottle, combine one cup of warm water with one cup of distilled white vinegar, available at any grocery store. Spray the solution directly onto the tub surface, and wipe clean with a dry cloth. This restores the shine and breaks up any soap scum that has developed. Repeat after each bath or as necessary to keep the tub looking its best.
Baking soda also works to clean and deodorize without damaging the acrylic tub finish. Again, in a spray bottle create a solution of two cups warm water and one cup baking soda. Spray this mixture directly onto the tub, and clean with a cloth or sponge.
For built-up soap scum, sprinkle a thin dusting of baking soda over the tub, and use a cloth to scrub the baking soda into the soap scum, breaking it up with ease. Rinse the tub with warm water to remove any leftover soap scum or baking soda residue, and repeat every few days or as necessary to keep your black acrylic sparkling at all times.
Rubbing alcohol is also effective at cleaning and disinfecting acrylic surfaces without causing damage. The combination of two cups water and one cup of isopropyl rubbing alcohol in a spray bottle works similar to the vinegar mixture and baking soda mixture noted above: spray the solution onto the tub, wipe with a cloth or sponge and rinse with warm water to remove any alcohol or soap scum residue. This method is safe enough to use daily, if needed.