404 
the members of the dential equation = 7 
+ —-- 
1 
Uap be the Bion efit off) andy 
that not Fe then when x becomes 
Se seh) St bers hs Hh and FCs) become F (4-8) 
art. 19.) or as Gis fincelns 
values of z, we must have 
ph pete 2)p'h4-H h. 
(e) tee + Hh 
are (2): 
and H 
be universally equal, we must 
H and p’'+H’ 
se: fs Seach of tea de: that is, 
 Sugtne Pond Ht’: 
aff(znb aa F(=)}. 
converse of this does not hold true ; 
for 22dx may be the ap of x*, or the fuxion of 
=*+4¢, ipo, He any constant quantity). So that in 
the equation dv = du, we may infer 
Let y=v Po, for the fluxional co-effi- 
cients of 0 cea when x becomes «+-/, 
obec ate iti te 
u becomes u+-p’h-4-H’h, 
Petts) 
therefore a A h+Hh) (u+p’h+H’h), that is 
vu-t(v pup) h 
RUA ied oun cotanmiead bel 
for vu; and Hh, 
@ terms that that contain eas 1 
HA, HA, he, me fare 
= h+-H" h. 2 
Hence it eerie ape te of 
therefore u 
pe pe pdz=dv, rac gap domes det 
therefore. 
dy=vdu+udv, 
Hence this rule, 
To find the fluxion of the peodact of two functions of 
the same variable quantity: Multiply each function by 
the fluxion of the other, and add the products. 
If we divide the two sides of the equation d(uv) = 
ve Serpe uv, we have (42) — pa 
Gitak which ‘on pany vendliy nd hie Mexia 
and 
amount of tl 
three 
of the product of any number of factors whatever. For, 
te ape ems 
MO. ds 
en on ; 
and therefore, 
d(st 
Aira Styl 
Inthe sie manner it ay be shew, that whatever 
be Se member always 
d(stuv... &c. du dv 
stuv... &e. a tH 4, be. 
‘aud inenen the Saxton of the Ht ne i 
stuv...&e.X% {45 +545 w+ &f. 
If now each term be actually multiplied by the 
FLUXIONS. 
stwv... &¢, the denominator will be taken 
away, and the result will ese us this rule’: 
To find the fluxion of 
and the sum of all these results will be the flaxion re- 
. 30, Next, let it be required to find the fuxion ofthe 
fraction y= ~. Because v=u y, by rule (F), dv= 
ydutudy ; put — Coe in the second sci 
ber, and we have d v= """ 4 wdy, sai bichon 
saves aaa 
dy= = " 
errnetee oh 
To find the fluxion of a fraction, a 
of the numerator by the the fnsion 
pesparkayrncsy ) 2 weap numerator; culbtrast ties lat: 
ee roduct from the former, and divide the remainder 
e square of the denominator. 
rn Let u be a function of a variable quantity x, and 
again let y be some function of u. It is required to in- 
te a general rule for finding the fluxion of y re- 
to x. 
be te Bese oe 
think en « becomes 2+ teh) Agila, 1d 
et 
uth, ui k for i Ih) 
i: co-efficient of relative be then, aus 
Vovonee ie of « becomes 
y+p’ Pipes , Rising «ana that vanishes when 
cat) 5 od, eee dos vteasin ob ack 
* yep (ph+H h)-£K(p i441) 
| Sy+pp h+(p A+ ney 
But as when cig vas pone gc consequently 
K=0, we soc Ett Bow the loot tema of ex- 
ression, (H”" being a quantity Becdigint ut oa 
=0), then it will that when « increases 
to «+A, +: oe unction of 2, increases to 
y+ Pip 
ter 
to ‘ai 
pee bat p beg ad therefore, relatively # 
v to xz, amd u; aw, 
rn No a pei a he ay 
en to u, Xe we 
have this rule 7 ; “ 
(1) 
To find the fluxion of y a function of u, which is 
itself a function of a variable quantity 2 ; find the 
fluxion of y considered as a function of u, without any 
toz: hinge japaalel rhe pd acre lh 
find the fluxion of u considered as a function of x, and 
ps2 mal aa end the speak will he spe fare 
Sate dave 
32. We now ive some exam 
ein dl thas aia ples 
Exampce 1. Let uaz 4/z, or, as it may be otherwise ex- 
pressed, 3 F By rule (A), d neat ee, 
x 
“Gn ee | 
Ex. 2, alc ab that is, u = a-'. In this cad, 
7 
Direct 
Method. — 
——— 
