Elements of Differential Calculus (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 Excerpt: ... a: . 2xz, 2.4a;5, 2.4.6X1. _ 4/1-2 3 3-5 3-5-7 From which, multiplying both members by 1--x2, we obtain 15. (i--jr) ' sin-1. = x H R DV ' 335357 in which, putting x = sin 6, we have sin20 2 sin40 2 4 sin60, 16. 0 cot 0 = 1 h It. 3 3 5 3 5 7 When the determination of the successive derivatives of a higher order is laborious, a simpler method may be employed provided the development off'(x) is known. Thus, since sin-1 x is an odd function, which vanishes with x, we assume f(x) = sin-1. = Ax + Bx + G5-j-Dx1 + etc. Differentiating, we have /'() =, = A + zBx + $Cx + iDx + etc. (1) /i--x Developing 1/ 4/1--x, we have, provided x 1, V ' 2 ' 2.4 ' 2.4. 6 ' 3.5.7...(-l)+ ( ' 2.. 4. 6... 2ft W or, placing Ir1 731 r X 6=j5" = A' = ' 30 = i?" etC-'-v v /v /y--v v-8 The coefficients represented by 2?, 2?3, 2?s, etc., are used in the higher branches of analysis, and are called Bernoulli's numbers, 128. Extension of Taylor's Formula to functions of two or more sums of two variables each. Let u=f(x, y), x and jy being independent variables; and let it be required to develop /(x--/iy y--k), in which h and k are variable increments of x and respectively. If, in/(#, y), x be increased by h, and/(#--h9y) be developed by Taylor's formula; then if, in each term of the result, y be increased by k and developed in a similar manner as a function of y + k, we shall have/(# + k, y + k) = sum of the latter developments. Otherwise, develop f(t) = f(x + ht, y+ kt) as a function of t, by Stirling's formula, and in the result make = 1. 0W = /( + M + ), .-0(0) = /(, j) =. In order to express conveniently the successive deriva tives with respect to /, place x + ht = ze/, and j +--giving 0(/) = /(-/, J). Hence ( 102), dcp(t) _dP(t)dw df(t)ds dt dw di ' ds df Su...