Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 153

subtract to obtain

AY = AX + Ax) - AX),

and then divide by Ax to obtain

(3.52) AY =f (x -I- Ax)- f (x)

Ax Ax

This quotient, which is clearly a quotient of differences that are calculated
in a special way, is called a difference quotient. Difference quotients have

already appeared in our problems, and we shall see later that they have
important interpretations. Leaving the hosts of applications to be par-
tially revealed later in this textbook, and to be continually revealed to
those who pursue further studies in the sciences (including mathematics),
we now come to one of the two most important ideas in the calculus. If

the difference quotient in (3.52) has a limit as Ax approaches zero, then
f is said to be differentiable at x and the limit is called the derivative of f
at x. In case the limit fails to exist, the function is said to be nondiffer-
entiable at x and we say that the derivative of f at x does not exist.
There are two very different and very useful notations for derivatives.
The first, appearing in the formula

(3.53) f'(x) = Jim Y= Jimf(x + Ax) - f(x),
&,,-.o Ax AX-0 Ax

is usually read "eff prime of ex," but it can be read "eff prime at x" or
"the derivative off at x." This "prime notation" is called the Newton
(1642-1727) notation.t The second notation, appearing in the formula

(3:54) dy Ay _ f (x + Ax) - f (x)

dx ,-.o x_ urn Ax '

is read "dee y dee x" or "the derivative of y with respect to x" and was
originated by Leibniz$ (1646-1716). There will be times in the future
when we will consider dy/dx to be the quotient of the two numbers dy
and dx. Meanwhile, the whole symbol dy/dx is to be regarded as a
single symbol, just as the symbol H represents a single letter of the alpha-
bet and not 11 divided by 11. A longer and perhaps dismal discussion of
this terminology and notation appears in a remark at the end of the prob-
lems of this section; congratulations can be bestowed upon readers wise
enough to know that the discussion is semisuperfluous.
According to an old and honorable tradition, the definition of dy/dx and
t The original Newton notation was the "dot notation" or the "flyspeck notation"
which employed j instead off', but replacing the dot by the prime is a clerical modification
that preserves the original idea of Newton.
$ Leibniz, like Newton, published his scientific works in Latin. The Latin spelling
"Leibnitz" is sometimes seen and sometimes helps people to pronounce the name correctly.