A Derived medium is one where we know its exact chemical composition. We can order chemicals and make such a medium. These are used in studies where exact knowledge of medium’s carbon source is required.
A complex medium has a chemical composition that is only partially known. It contains materials such as yeast extract, soybean extract, or meat extract. We do not know the exact chemical composition of some of these extracts. Complex media are often easier to use.
A differential medium is one that permits growth of some microorganisms but not others. There are many examples of this. Acid may be added to a medium to allow for the growth of fungi and to prevent the growth of bacteria. Potato Dextrose Agar is a medium used for this purpose. Bile may be added to a media to encourage the growth of one bacterium over all others.
Violet Red Bile Agar is an example of this.
Note: bacterium is singular; the word, bacteria, is plural.