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

388

RULES FOR FINDING f '(x).

Rules VII. 71 This Rule is of great use in finding the derivatives
duced from of inverse functions, and by means of it may deduce Rules VII,
vlI. Ix. X, XI, from Rules VI, VIII, IX, respectively, as follows:
by means of
Rule XII. If x = logay = +((y) suppose, then y = a" =f(x) suppose; and therefore, since
f'(x)= (p- and p'(y) = oga by Rule VI,
(y) log a y
we have f'(x) = log a. y = log a a"; which is Rule VII.
If a = sin y = ( (y) suppose, then y = sin-' a =f(x) suppose; and therefore, since
1 1
f () = - = -c by Rule VIII,, ( Coy
we have f'(a) = 2, since cosy V/1 _- 2; which is
2/1-,2
Rule X.
If x = tan y = p (y) suppose, then y = tan-' = f(x)
suppose; and we have therefore
f'($) =, = Cos2y, by Rule IX. which = 1 a, since
fl( (Y) 1 +.i
tany = x; which is Rule XI.
Rule XII. 72. We may also by this Rule obtain the following
thlsedtof derivatives which it is sometimes useful to remember.
sec-' x
vers-'1 If x = sec y = / (y), then y = sec-' x =f(x), and we have
therefore
f '(c) == _ = osY by Cor. Rule VIII.= 1
qv(a? ) sin y -1
1
since cos y =
Hence, if f(x) = sec- x, f' (x) =
x -/x - 1
Again, if a = vers y = (y), then y =vers -' = (),
and we have therefore