Page 117 - Foundations Of Differential Calculus

100 6. On the Differentiation of Transcendental Functions
180. We are investigating the differential of the hyperbolic logarithm of x
and we let y =ln x, so that we have to define the value of dy. We substitute
I
x + dx for x so that y is transformed into y = y + dy. From this we have

dx
y + dy =ln (x + dx) , dy =ln (x + dx) − ln x =ln 1+ .
x
1
But we have seen before that the hyperbolic logarithm of this kind of
expression 1 + z can be expressed in an infinite series as follows:
2 3 4
z z z
ln (1 + z)= z − + − + ··· .
2 3 4
When we substitute dx/x for z we obtain
dx dx 2 dx 3
dy = − + − ··· .
x 2x 2 3x 3
Since all of the terms of this series vanish in the presence of the first term,
we have
dx
d ln x = dy = .
x
It follows that the differential of any logarithm whatsoever that has the
ratio to the hyperbolic logarithm of n : 1, has the form n dx/x.
181. Therefore, if ln p for any function p of x is given, by the same argu-
ment, we see that its differential will be dp/p. Hence, in order to find the
differential of any logarithm we have the following rule:
For any quantity p whose logarithm is proposed, we take the differential
of that quantity p and divide by the quantity p itself in order to obtain the
desired differential of the logarithm.
This same rule follows from the form
0 0
p − 1
,
0
2
to which we reduced the logarithm of p in the previous book. Let ω =0,
ω
and since ln p =(p − 1)/ω,wehave
1 ω ω−1 dp
d ln p = d p = p dp = ,
ω p
since ω = 0. It is to be noted, however, that dp/p is the differential of the
hyperbolic logarithm of p, so that if the common logarithm of p is desired,
this differential dp/p must be multiplied by the number 0.43429448 ... .

1 Introduction, Book I, Chapter VII.
2 Introduction, Book I, Chapter VII.