An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

CHAPTER VI.
THE DIFFERENTIAL NOTATION.
80. WE have hitherto adopted Lagrange's notation,
f(X) -f( )
f'(x), to represent the limiting value of,f() when
x' approaches x; this notation is often very convenient, and
is perhaps more easily understood than any other, at least
when the student enters upon this subject for the first time;
and this is the reason why we have used it in obtaining
the preceding Rules. But there is a far more elegant and
powerful notation, due to Leibnitz, called the differential
notation, which we now proceed to explain; not however for
the purpose of abandoning the former notation, for we shall
often make use of it, as it is preferable in certain cases to
any other.
Theratio 81 In the differential notation instead of representing
of the sym
bols d-f (x)
and dX the limiting ratio of f(- when.' approaches,, by a
taken to O- - b
epresent single quantity, such as f' (), we represent it by the ratio of
two arbitrary quantities denoted by the symbols df(x) and
db, which, for a reason we shall explain, are called the
differentials of f () and v; the letter d being simply an
abbreviation of the words "differential of."
Definition We define df(x) and dX as follows:
of df(x)
and dx. df(x)
df(x) and dx are two quantities whose ratio d
dx
is equal to the limiting value of (x- () when x' apX -x
proaches x. Or more simply; df (x) and d x are two quantities
such that df(x) = f(x).
such that =d xW
dx