FLvxloas, direct method of, the doctrine of this part of fluxions is comprised in these rules.
1. To find the fluxion of any simple va riable quantity. the rule is to place a dot over it : thus, the fluxion of x is X, and of .y,.y. Again, the fluxion of the com pound quantity x±y, is ; also the fluxion of n—y, is X---y 2 To find the fluxion of any given power of a variable quantity, multiply the fluxion of the root by the exponent of the power, and the product by that power of the same root, whose exponent is less by unity than the given exponent This rule is expressed more briefly, in algebraical characters, by n x S.7 = the fluxion of 11 x Thus the fluxion of x3 is •x 3 X xi = .3 .v= ; and the fluxion of xs is x. X 5 X .x+ = 5 .14 x. In the same manner the fluxion of is 7 Y X ; for the quantity a being constant, y is the true fluxion of the root 0 y Again, the flux ion of will be Q X 2 z i x for here x being put = we have. = 2 z i; and therefore xt.'• for the fluxion of x (or a'-1-z' -4) is= 3 z V a' 3 To find the fluxion of the product of several variable quantities, multiply the fluxion of each by .he product of the rest of the quantities; and the sum of the pro ducts, thus arising, will be the fluxio.n sought. Thus, the fluxion of x y is X y. r; that of x z is Xy 2+ ifxz±±..cy; and that oftyxyzistlxyz-Fi:vyzi Yvxz-l-ivxy. Again, the fluxion of a+xxb—y • 4. To find the fluxion of a fraction, the rule is, from the fluxion of the numerator, multiplied by the denominator, subtract the fluxion of the denominator multiplied by the num erator,and divide the remainder by the square of the denominator. Thus, the fluxion of! is y x ; that of Y' x , x+ y and that of or 1 x 1' x y + z , is X x and x y x+ SO of' others.
In the examples hitherto given, each is resolved by its own particular rule ; but in those that follow, the use of two or more of the above rules is requisite: thus (by rule 2 and 3) the fluxion of x7 y' is found to be 2 yp + 2 x that of x7 -, is found (by rule 2 and 4) to be Y' and that of !LE, (by rule 2, 3, and 4,) found to be x.i X z— s' ya
5. When the proposed quantityis affect ed by a coefficient, or constant multiplica tor, the fluxion found as above must be multiplied by that coefficient or multipli cator: thus• the fluxion of 5 x3, is 15 s' for the fluxion of x3 is 3 s, tiplied by 5, gives 15 X. And, in the very same manner, the fluxion of a xn will be n a xn-1 Hence it appears, that whether the root be a simple or a compound quantity, the fluxion of any power of it is found by the following general Rule: Multiply by the index, diminish the in dex by unity, and multiply by the fluxion of the root. Thus the fluxion of 8 it 1: the fluxion of 4 = 24 x5 2'.! and 12 .
the fluxion z 4 = 20 z — i 3 3i 5 Having explained the manner of de termining the first fluxions of variable quantities, it is unnecessary in a work of this kind to enter upon the second, third, &c. fluxions, we shall therefore proceed to FLuxioNs, inverse method of, or the manner of determining the fluents of given fluxions.
If what is already delivered, concerning the direct method, he duly considered, there will be no gret difficulty in con ceiving the reasons of the inverse method; though the difficulties that occur in this last part, upon another account, are in deed vastly great. It is an easy matter, or not impossible at most, to find the fluxion of any flowing quantity whatever; but, in the inverse method, the case is quite otherwise; for, as there is no me thod for deducing the fluent from the fluxion by a direct investigation, I so it is to lay down rules for any other forms of fluxions than those particular ones, that we know, from the direct method, belong to such and such kinds of flowing quantities ; thus, for ex ample, the fluent of 2 x d is known to be x'; because, by the direct method, the fluxion of is found to be 2 x I:: but the fluent of y X is unknown, since no expres sion has been discovered that produces y for its fluxion. Be this as it will, the following rules are those used by the best mathematicians, for finding the fluents of given fluxions.