DRiavere adz 1 yas a 
oats, Jay EV at 
edz a. 12 
V¥Ui—2) =—($+75)v0—*) $6 
ay = BV (0 2) 4} ae (in. = 2) bo. 
wtdz 1.32 
x3 
Wins =~ Gta = 
+ st arc (sin, =r) 4c. 
131. However, if n were negative, formule (E) and 
, and (F) would no longer apply ; but, by making z = 
1 
? ‘>? we find, 
dz pe —2-1dz 
FVI—#) ~ V(#—1)’ 
dx 1 dz 
2/(@= 1) Jase 
Besides, we may find formule which shall apply direct- 
ly, by ing as in last article ; for, taking the flux« 
ion of z*+14/(1—a*), we get 
7 dz —__ x(i—=*) 
wr/(L— 2?) ~ — (n—=1) 2 } (G) 
n—2 de 
+71 J 0—) 
When z is an odd number, this formula makes the 
dx 
fluent ESE i 
at last depend oe fa which, by Ex. 
2. art. 33, is 
LI Vv0+2)4+V0—2) 
ooh d Saeahiaet os) 
se ftvC—-)} 
In like manner, we find 
adzx 
gm! 
VQae—2*)—~ om anes t (H) 
(2m—1)af* 2—dz 
+ m J/(2aa—2) 
Of Exponential Functions. 
132. It appears, from art. 26, rule (C), that 
pruzty 
L(@) 
Let V be any algebraic function of ar, then, because 
oa? 
d (a*) 
‘ _ dr= » if we put a*° =u, we have Vd = 
a* 1. ( 
Vdu “) ’ . % 
it ive a.., 
algebraic form, For example, let V= ——-“——_ 
Vl + a") 
pas ade du 
pe! V+ el Ee: 
uent of this last i 
F : rad t expression may be found by the 
Let z be any function of x; then, e being the num- 
ber of which Nap. log. is unity, we have d (z¢*)= edz 
+ zda; , ; 
FLUXIONS. 
Judizut—fidy, making u = 2", 
443 
Inverse 
Method- 
—_—— 
dz - 
fe ds (: +7 )=e fe. 
a. 
For example, let z = «3 —1; then = 32", and 
zr 
Sede (3 2? 423 — 1) =e" (a3 —1)+-<¢, 
133. In other cases, we may have recourse to the 
method of integrating by parts, (art. 114. Rule IV.) 
Thus, let the fluxion be «* dz.a*; then, by the formula 
dtz=a‘dz, we 
ara” n 
fedeaa L(@) ~E(a) vd e. A 
By treating a* x*—'d x in the same manner, and re- 
peating this as often as isnecessary, we find Ja ade 
{ a na! n(n—1) a-* 
* UL(a) ~ TF (a) t~ 13a) 
1.2.3...” 
~~ Seat + ¢. 
In this series, 1.* (a), 1.3 (a), &c. mean the square, the 
wet Te the Be t n is negative, by following th 
134, e n isn ve, ‘ollowin 2 
same method, we may anordiee the fete Bo: of x, Tae. 
cordingly, from the formula fu d ¢ ='u t — ft du, ma- 
king u =a", and dt =<, we find 
aad x 
ard x —a l(a) 
G te =G—pe ta) =r 
By repeating this transformation, we bring the fluent 
ad d. 
J {Sto depend on <=. This last fluent has long 
exercised the ingenuity of analysts in endeavouring to 
reduce it to circular ares or logarithms, but without 
success. It to be a transcendental of a pecu< 
liar kind. For want of a ri s method, we may 
employ a series ; thus, putting A for l(a), we have, by 
art. 53, ‘ 
A®x? A3z3 
@=l+Ar+ —3-+ 95 
Therefore, multiplying by and taking the fluents, 
we find pad he 
A?x? ~ Ai x3 
L(2)+A a + ga t+aagte++ +e 
135. If n be a fraction, either of the preceding me« 
thods will serve to reduce the exponent of x to some 
fraction between 0 and +1 or — 1; and then, recourse 
may be had to the method of infinite series, which we 
are afterwards to explain. 
regard to the fluxion of 
zdx.a*, will ly equally to zdx.a", supposing z to 
ti say kina eee act: 
+ &e. 
Of Logarithmic Functions. 
Let it now be 
zdx1."(2x), putting 1." (2) to denote the nth power rithmic 
of the Napierean log. of xz, and supposing z to be any y=) 
braic function of zx, 
n be a positive integer, then recurring to the 
required to find the fluent of of loga- 
