. 1820.] of Taylor's Theorem, S;c. 101 



by substituting in the general theorem and writing A x for d x, 

 we get A {x (x + A x)} = 2 A .r (x + A x) ; (d ,i constant.) 



Example 6. — To find the incremedt of .r (.i + A x) {x 4- 2 A x). 



In order to simphfy, put w = A .r, a constant quantity ; then 

 will d {x {x 4- A x) (.1- + 2 A x) = d {.i'^ + 3 w x- + 2 w^ x) 

 — 3 IV x"' + 6w' X + 2 io\ d (3 lo x^ -f 6 iv^ x) = 6 w'' x + 

 6 w^, and d (6 ?<;'■' x) = 6 i<;3. Then by substituting in the gene- 

 ral theorem, we have the required increment ; viz. 3 w x- + 

 9 id'^ X -\- Q 10^ =■ 3 w (x + lo) (j: + 2 w), by restoring A x, we 

 get A{a:(x + Ax) (x + 2 Ax)} = 3 Ax{(x + Ax) (x + 2 A x)}. 



Example 7.— To find the increment of x (x -|- Ax) (x + 2 Ax) 

 (x + 3 A x). 



Put A X = 10, a constant quantity ; then will the above 

 expression become x^ + Q lo x^ + 11 w'^ x'^ + 6 w^ x, then will 

 dix' + Qivx' + \\iv"x''- +Gwx)= 4M;x3 + 18?<;-.i^ + 22w^r+ 6w* 



d(,'^wx' + l%io"x"- + 2-2io\v)^2 = + Gvrx- + \^w\v+U%o^ 



d(\2 lu'' X- + 3Q luKi) ^ Q. . . = + 4?t7'.r+ 6«;* 



</(24 26-^0 -H 24 = + w' 



The required increment .... =.Aiox^ +2Aw°x- -\- AAiv^ x -\-2Aio* = 



4 ti; (x3 + 6 if7 x^ + 1 1 ?o- X + 6 70') = Aw Ux + ?c) (,r + 2 lo) ( x + 3 «;)) = 



4Axf(x + Ax) (x + 2 Ax) (x+ 3 A .r)\ Hence we conclude 



that A ^x (x + A x) (x + 2 A x) (x + « A x)j= (« + 1) A x 



((x + A x) (x + 2 A x) (x + « A x)). 



Example 8. — To find the increment of-. 

 Put If = (^ X = A X, a constant quantity, then will 

 d(^-^ = d{x-') = - w x-^ = --^^^ d {- 10 x-^) H- 2 = 



vf- x-^ = 4 ; ^ (2 w'^ -i.-') ^ Q = - xo' X-' = - '^ ■ ^c. 



Therefore A f-) = - Ji + -^ - -^' 4- &c. = -^ ^ = - 



10 A jr 



Example 9. — To find the increment of — j. 

 Put M) = rfx = A X, a constant quantity, then will 

 d (jj = d (x-0 = - 2 «; X-' = - if ; rf (- 2 it- x"^) -- 2 



= 3 w X-* = '—, d (G 10" x-") -^ 6 = — 4 ?<•■' x-^ = — 



— -; &c. Therefore A (— = t + —, - + 8cc. = 



(_2__^ J J_ '2 10 X +' tc* 2 X \ X + A" 



X + id) X* ~ *J (X + it)' ~ ~ x\x + Ax)*' 



