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

50

THE DIFFERENTIAL NOTATION.

triangle POQ has any existence, and the erroneous supposition that x +d dx is equal to 2x.
We shall endeavour however to shew hereafter, by means
of the principles already established, that this method of
using differentials must always lead to correct results, certain precautions being taken; and this is important to shew,
since this method is extensively used, and indeed must be
used in many cases to avoid complexity. At present we shall
say no more upon this subject, except just to remark, that
df(v)
this is the manner in which the notation d( has come
do
f(d) _ f(X\
to represent the limiting value of f( - ) when v' apX — X
proaches x: for it is easy to see, by the example just given,
that dfv(, simplified by neglecting the differentials df(v)
and dx compared with ordinary quantities, will always be the
same thing as this limiting value.
We adopt 86. It is important to observe here, that we adopt
the notation
f( and only the notation f(x) and the term Differential. We do
the term d v
differential, not define df(x) and dx to be what Sf(x) and Sxv become
but not the
notion that when in a state of extreme smallness; all that we say is
d f (x) and
dx are in- this, that df(x) and dx are two quantities, be they small
finitesimals. or large, whose ratio is equal to fJ (x), i.e. the limiting value
of f() or f () - ) when (' approaches x.
BO XI - O
Geometri- 87. In the example just considered, if we produce PO
cal representation of to any point U and draw US perpendicular to PU to meet
the quanti- SPT in S; then
ties df (x)
and dRx.
= tan PTS = lim. val. of7 f() when I' app. x,
PU i -iv
= f (f);
hence SU and PU are the differentials of f(x) and x, since
they are two quantities whose ratio =f'(x); we have therefore SU - df(x), PU = dw.