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THE CONCEPT AND DEFINITION OF A DERIVATIVE

Concepts that shape the course of mathematics are few and far between.                              The derivative, the

fundamental element of the di?erential calculus, is such a concept. That branch of mathematics called

analysis, of which advanced calculus is a part, is the end result. There were two problems that led to the

discovery of the derivative. The older one of de?ning and representing the tangent line to a curve at one

of its points had concerned early Greek philosophers. The other problem of representing the instanta-

neous velocity of an object whose motion was not constant was much more a problem of the seventeenth

century. At the end of that century, these problems and their relationship were resolved. As is usually

the case, many mathematicians contributed, but it was Isaac Newton and Gottfried Wilhelm Leibniz

who independently put together organized bodies of thought upon which others could build.

The tangent problem provides a visual interpretation of the derivative and can be brought to mind

no matter what the complexity of a particular application. It leads to the de?nition of the derivative as

the limit of a di?erence quotient in the following way.                (See Fig. 4-1.)