Application of the Flnxional Ratio, SfC. 309 



II. To find the fluxion of (x^-\-x^)'^ ; in this example D"x" = 



nx' 4(2xx'-\-3x^x') ^ ^*' , 



{x'+x'y,^nd-=: ^; hence D"x"X--(x^+3;3)* 



X^ ,-7— r— =4(z=^+x3)=»X(2zx- + 3x=a;-). 



III. To find the fluxion of {ax — xxY; here DV=(az — xx) , and 



nx- i(ax-~2xx') , -r^ „ ^2;- 1 J(«x'— 2xx') 



— =^^ -> hence D''a;"X — = (aa;-xx)^X-^-- = 



X ax — XX X ^ ^ ax — XX 



Ox' — 2xx* 



2(ax-xx)* 



IV. To find the fluxion of j^=a{a' -y^Y^; here D''a;'' = 



{a^^-r^f 



a(a^-2/^) ^and— =-^j^-^,henceDVX — =a(a='-y=') ^X 

 -^X-2yy ^ ayy 



The fluxion of r is a(a=^— 3/ *) '^ X ratio 1 — -; = 



bay'y 



2{a''—y'Y 



A . . IXr a;- 



The fluxion of - is x"' X ratio — = — — • 



X X x^ 



1 1 3a;- 3x- 



The fluxion of — r is — r^ X ratio - -— = - — — • 

 ax^ ax ^ X ax* 



3 3 —ax* 

 The fluxion of ¥ax — a^Y is |(ax - aM^ x ratio —^ 7= 



a[ax — a^)x'. 



The fluxion of — rT(a^'" + «')""^' 's — — rrx^+aM""^' X ratio 



^ J, — = (x"'+a=')"Xmx'"-ix-. 



X +0-^ 



(a'"+2r'")"+i (a'"+z"')''+i . (n+lW^-'z- 

 The fluxion of ^ — j- , \. is ^ , , ,. X ratio „;. „ 



= (a'"+;s'")"X2r'""'2r'. 



When the fluent is the product of two or more flowing quantities, 

 as xyz, the fluxion of each is to be taken seriatim, the other remain- 



VoL. XXIV.~No. 2. 40 



