Page 116 - Foundations Of Differential Calculus

6

On the Differentiation of
Transcendental Functions

178. Besides the infinite class of transcendental, or nonalgebraic, quanti-
ties that integral calculus supplies in abundance, in Introduction to Analysis
of the Infinite we were able to gain some knowledge of more usual quan-
tities of this kind, namely, logarithms and circular arcs. In that work we
explained the nature of these quantities so clearly that they could be used
in calculation with almost the same facility as algebraic quantities. In this
chapter we will investigate the differentials of these quantities in order that
their character and properties can be even more clearly understood. With
this understanding, a portal will be opened up into integral calculus, which
is the principal source of these transcendental quantities.

179. We begin with logarithmic quantities, that is, functions of x that,
besides algebraic expressions, also involve logarithms of x or any functions
of logarithms of x. Since algebraic quantities no longer are a problem, the
whole difficulty in finding differentials of these quantities lies in discovering
the differential of any logarithm itself. There are many kinds of logarithms,
which differ from each other only by a constant multiple. Here we will
consider in particular the hyperbolic, or natural, logarithm, since the others
can easily be found from this one. If the natural logarithm of the function
p is signified by ln p, then the logarithm with a different base of the same
function p will be m ln p where m is a number that relates logarithms with
this base to the hyperbolic logarithms. For this reason ln p will always
indicate the hyperbolic logarithm of p.