Page 74 - Schaum's Outline of Theory and Problems of Advanced Calculus

Derivatives

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 differential 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 defining 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 definition of the derivative as
the limit of a difference quotient in the following way. (See Fig. 4-1.)

Fig. 4-1

Let P o ðx 0 Þ be a point on the graph of y ¼ f ðxÞ. Let PðxÞ be a nearby point on this same graph of the
function f . Then the line through these two points is called a secant line. Its slope, m s ,is the difference
quotient

y
f ðxÞ f ðx 0 Þ
m s ¼ ¼
x x 0 x
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