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 DIFFERENTIATION. 53
f'(x) =,'() by Rule XII. Hence, whichever way we prof (y)
ceed, the value of obtained is the same.
do
91. Rule XIII. shews that if we find dy in terms of dv RuleXIII.
from the equation y= 0((v), and dv in terms of dx from the
equation v = +(x), and so dy in terms of dx; the value of
thus found is the proper value; i. e. the same value that
dx
would be obtained if we found y in terms of x directly by
eliminating v, and then differentiated. For by differentiating
the equations y = (p(v), v = f (x), we obtain
dy = p'(v)dv, dv = '(x)dx, and.'. dy- = () () ().
Now if f(,) be the value of y found directly in terms of a, we
have - =f'(x), which is the same as the former value, since
dx
f'(x) = '(v) +'(x) by Rule XIII. Hence, whichever way we
dy
proceed, the value of - obtained is the same.
dx
92. If we use the differential notation, this Rule does RulesXII
not appear to want any proof; for it is stated in that no- and XIII
seem to
tation, as follows. If y be a function of v, and v a func-wantno
proof in the
ddy dy dv.... differential
tion of x, then - - -. Now, since dv divides out in notation.
dx dv dx Why
the second member, this appears to be self-evident. But we less they
dy ought to be
must remember that the - in the first member is supposed to proved.
dov
be derived from y expressed directly in terms of x; whereas the
dy
d- obtained from the second member, by dividing out dv, is
dox
not derived from y in this manner; and though it is easy to
see that the result is the same in both cases, yet it ought to be
formally proved for the sake of exactness.
The same may be said of the preceding Rule also.