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

RULES FOR FINDING f' (X).

43

ant of d? or not, we may always put v' in place of jfshould be
f (X'y) -f~xy) independwe please, and so find f(x'y), and thence ant of each
iv - iv other.
and then, by the previous Rules, we may determine the
limiting value of this quantity when x' approaches x. And
the same may be said of f'(xy). The following is the Rule
for which it is necessary to make use of this notation or
something equivalent to it.
t t 78. If we are given a relation between y and Rule XV.
two other variables, u and v which are functions of x;
i. e. if
Sr (UV), u= U f(X), v = X(x),
in which case y will be some function of x, f (x) suppose;
then
f' (x) = /i' (uv) Wfr' (x) + /i (U v) x' (x).
f(') -f () _/ (u' V) - /(UV') q5 (uV') - qp (u v)
For +,
x Ml X X x - x
and 1PKu'v')-5(u'v') _ ~(U'V) -(UV'). 'kk (Wv) - xJ (iv
X X UUX - xU
~ (UV) - (Uv) - (UV') - - (Uv) x (I') X (x)
iv-iv Llv -V X -(
Now when x' approaches x, the limiting values of
J(v ') (,~v) ((UV')- 4 (Uv) x _ _ x (V)
and
are respectively x'(w), (P' (u v), and X'(x). Also since v'
becomes v when Iv' iv, the limiting value of
u-u
* If we suppose for a moment that v' is not x (x') but some constant, then
p'(uQv') is the limiting value of when x' approaches x: there~ ~~~~p~~~~ (~u whe )' ppoacesavthre
fore q)U'v' P(UV ') may be diminished ad libitumn by sufficiently
diminishing x' x; and this is evidently true whatever be the value of V; therefore it must be true if we suppose v' to vary in any manner while we diminish
-x; it is therefore true if v'=X(x'), and therefore by Lemma VI. Cor. 2,
'(Vuv), which is the limiting value of pV'(? v'), is also that of ~(u'')-, (uv')
supposing v' = x (x').