Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle, though analysis as a formal concept is a relatively recent development. As a formal concept, the method has variously been ascribed to Ibn al-Haytham, Descartes (Discourse on the Method), Galileo, and Newton, as a practical method of physical discovery.
In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position).
^Differential calculus, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.