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