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

32.

RULES FOR FINDING f' (X).

fo. (x) f6) 95(x)
Cor.1. 61. If,(x) then
f (X)
0s (X)
f'(x)-)-9()(x)
For then f(x) + Qx) = 5(x(); and, as in the rule just
proved, we have
f' (X) 4' (0) + f W) 4" (x) = 9'(u);
whence f'(a,) = ( f x4"
4' (X)
)' Qe) N'" (x). - ()) 4" (X)
putting for f(x) its value.
Rule V. 62. If f(x) =xX, then f'(x) = nx': '; whether n be
f integral or fractional, positive or negative.
f (,V) (~n>~" m
For then f(x' f x"
X - X (V X- L
nn-
_ Z if we put x'= vz.
z-1l
Now X' x when z = I, therefore by Lemma II. the limiting value of the second member of this equation when z
approaches 1 is the same thing as that of the first member
when x' approaches x; but by Lemma XI. n is the limiting
z'n - i
value of 1 when z approaches 1. Hence f'(x) = nmv".
Examples 63. By means of these five Rules we may find the deriof these vative of any rational function of x. For example:
five rules.
Let f (X) = aoe+ aix + a2IV......a.,I
then by Rule III. f'(x) is found by taking the derivative of
each term separately; also by Rule I. the derivative of a,
is 0, and by Rule 1I. the derivative of any other term a C"'
is aMl x the. derivative of x"', or aM mrn lg by Rule V.