An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

LIMITS

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9. Solve the first fifteen or more examples at page 50, Todhunter's
Differential Calculus, by Fermat's method, indicated by
Lt f(x + Az)-f(x)*
h- - o As
20. The Differential Calculus. We have seen how, by applying the formula f(X+ — /f( and, after certain algebraic
reductions, passing to the limit of the fraction, we have been
able to find the first differential coefficient of J(x). This process is called differentiation; and when a function has been thus
operated upon, it is said to be "differentiated." Now as there
are but six fundamental algebraic operations and only a few
transcendental functions in common use, we ought to be able
to devise simple rules and corresponding formule for writing
out the differential coefficient of any such given form. For
example, in our last lesson we found the derivative of x4, n being
a positive integer, to be inx'1. It can be shown that this result
holds for all values of n; consequently we have the rule for
differentiating any power of a variable with respect to that
variable: "Take the product of the exponent and te ivariable
with its exponent diminished by unity." Thus: -xd = 7 6,
dx dx
These rules and the formulae expressing them in mathematical symbols constitute the differential calculus. Hence,
by means of a few simple rules we are able to tell what the
* Strictly speaking, Fermat's method is one for determining the maxima and
minima of functions of a single variable; but as it is so closely related to the fundamental formula, f(x + t) - f(x), that Lagrange and other eminent mathematicians
would have given him credit for the invention of the differential calculus, it seems
quite proper to associate his name with the general formula of differentiation.
t Invented by Isaac Newton (I642-I727) and Gottfried Wilhelm Leibnitz (I646 -I716) independently. While Newton certainly had priority, yet the notation of
Leibnitz was as certainly superior; and thus it is that the honors are quite evenly
divided.