How do I apply for a patent for an invention that is based on an already patented technology?

That is a difficult question to answer in a generic manner, as all "inventions" depend on the very specific facts of the particular development. Generally, to be patentable in the chemical arts, the progress must be novel and unobvious. Effectively, the result obtained should be new and unexpected. However, if the result was predictable but the benefit from the result greater than expected it could also qualify as unobvious. Some predictable but difficult to obtain results are also patentable. There are many other basis by which a chemical advancement may qualify for patent protection but it would be necessary to discuss specifics before an evaluation could be made.

In the mechanical areas, you simply have to have a feature which is novel and unobvious. Unobviousness, however, is a difficult concept to truly appreciate - not all examiners really understand unobviousness, and not all attorneys truly understand unobviousness. However, the bottom line is, is that if your invention is an improvement over existing patented technology, and is novel and unobvious, you should be able to obtain a patent. Most patents are based upon previous technology which is patented, so there is always a good chance that you can still obtain a patent - it just has to be new and unobvious.