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

OF IMPLICIT FUNCTIONS.

121

an equation
y and proceed as before. The following example will ex- beween
d x x and y.
plain the process.
Given y' - 4a2'xy + x4 =.., (1) to find what values of,x make y a maximum or minimum.
We have, differentiating
dx
d y x3 - a y
and. - =
dx a2x - y3
3
y = - makes this zero, (at least if x be not zero, for
dy o
then dx = o a case which we shall consider presently);
X3
now if y= we have in virtue of (1)
a?'
X12 X4
a- - 4a2 - + r4 = o or x8 - 3a8 ==, dividing out x4;
a a
or x = 3$a, and.. y = 3Wa;
dy
these values put in the denominator of d make it negative;
therefore by (165) we may put
dy
dy- ay - x3;
dx
dy 2= ady
-a — 3x
dov dx
negative, when we put x= 38a and.-. 0;
dx.'. = 38 a makes y a maximum.
Next, as to the value x=0 (and.-. y=o) we find, putting
dy
y =u as in 158 and dividing out V2, that u0= o, and..- =0
dcz