4-PrincipCalculus Page 107 Friday, March 12, 2004 12:39 PM
Principles of Calculus 107The derivative of a function represents its instantaneous rate of change.
If the function f is differentiable for all x ∈ (a,b), then we say that f is
differentiable in the open interval (a,b).
Introduced by Leibnitz, the notation dy/dx has proved useful; it sug-
gests that the derivative is the ratio between two infinitesimal quantities
and that calculations can be performed with infinitesimal quantities as
well as with discrete quantities. When first invented, calculus was
thought of as the “calculus of infinitesimal quantities” and was there-
fore called “infinitesimal calculus.” Only at the end of the nineteenth
century was calculus given a sound logical basis with the notion of the
limit.^5 The infinitesimal notation remained, however, as a useful
mechanical device to perform calculations. The danger in using the
infinitesimal notation and computing with infinitesimal quantities is
that limits might not exist. Should this be the case, the notation would
be meaningless.
In fact, not all functions are differentiable; that is to say, not all
functions possess a derivative. A function might be differentiable in
some domain and not in others or be differentiable in a given domain
with the exception of a few singular points. A prerequisite for a function
to be differentiable at a point x is that it is continuous at the point.
However, continuity is not sufficient to ensure differentiability. This
can be easily illustrated. Consider the Cartesian plot of a function f.
Derivatives have a simple geometric interpretation: The value of the
derivative of f at a point x equals the angular coefficient of the tangent
of its plot in the same point (see Exhibit 4.5). A continuous function
does not make jumps, while a differentiable function does not change
direction by discrete amounts (i.e., it does not have cusps). A function
can be continuous but not differentiable at some points. For example,
the function y = x at x = 0 is continuous but not differentiable. How-
ever, there are examples of functions that defy visual intuition; in fact, it
is possible to demonstrate that there are functions that are continuous
in a given interval but never differentiable. One such example is the
path of a Brownian motion which we will discuss in Chapter 8.Commonly Used Rules for Computing Derivatives
There are rules for computing derivatives. These rules are mechanical rules
that apply provided that all derivatives exist. The proofs are provided in all
standard calculus books. The basic rules are:(^5) In the 1970s the mathematician Abraham Robinson reintroduced on a sound logi-
cal basis the notion of infinitesimal quantities as the basis of a generalized calculus
called “nonstandard analysis.” See Abraham Robinson, Non-Standard Analysis
(Princeton, NJ: Princeton University Press, 1996).