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

44

RULES FOR FINDING f (X).

(h(u'v) -(UV)
is evidently the same thing as that of (u') - u ) which
is 0' (uv). Hence we have
f' (x) = )' (u ) ' (x) + ' (u ) X (V)
Examples. Let q/ (fuv) = U,
then q'(.uv) = vu"'- and op'(uv) =log u uv;
f-. f' () = v -L1 U /I (X) + log u " X' (V).
Suppose that u = xv and v = x, and. '.,'() = 1 X'(t)=1;
then f' (s) = x + logx x = vx (1 + log x).
Again, let c (uv) = u2 + v2 - uv;
then b'Y(uv) = 2u - v c'(uV) =2v - u;.'. f' () = (2u - v) #'(i) + (2v - u) X (,).
Cor. If y = (uvw) w being another function of x, i (X),
suppose then by putting f(x') -f() in the form
{c(Zu'V'w') - ( (uv'w')} + { (cuv'w') - 5 (uvw')}
+ [/(uvw') - (uvw)}.
We may shew exactly as before, that
f () = -' ( uvw) ' (a) + p'(u) W) X'() + cP'(uv w) [ (W).
This Rule we may evidently extend to the case where y is
a function of any number of functions of v.
Rule XVI. tt79. If an equation be given between x and y, which
of course makes y a function of x, we may find the derivative of y by means of Rule XV.
For, let the given equation be
+ (yx) = o;
in virtue of this equation y = some function of x, + (x) suppose; and 0 (yv), by substituting for y this value, becomes
also a function of x, f (M) suppose: then, by Rule XV, putting
y for u, and v for v, and therefore X'(v) =, we have
f ()) = ' (x) +' (X) + p' (y.).