what are the derivatives usually used in the preparation of bath soap or detergent?

Answer 1

Many soaps are mixtures of sodium (soda) or potassium (potash) salts of fatty acids which can be derived from oils or fats by reacting them with an alkali (such as sodium or potassium hydroxide) at 80°–100 °C in a process known as saponification. The fats are hydrolyzed by the base, yielding glycerol and crude soap. Historically, the alkali used was potassium made from the deliberate burning of vegetation such as bracken, or from wood ashes.

Soap is derived from either oils or fats. Sodium tallowate, a common ingredient in many soaps, is in fact derived from rendered beef fat. Soap can also be made of vegetable oils, such as olive oil. Soap made entirely from such oils, or nearly so, is called castile soap. The use of the word "soap" has become such a household name that even cleaning solutions for the body that don't have soap in the ingredients are referred to as soap.