-1

dcos" x=(-1)" sin x. sin cos x. sin cos2 x...sin cos2 -1x. dlog,"x=x-1. (log, x)-1. (log, x)-1... (log -1x) -1.

n-1

DEVELOPMENT OF FUNCTIONS,

(8.) d, f(x+y)=dyf(x+y).

δ

Taylor's Theorem. Let E,(u) represent what u becomes when + Da is substituted for a, then

If the n first terms of this series be taken, the limits of

(Dx)n

error are the greatest and least values of dru 1.2.3 n

...

provided there be no value of x between a, and + Do that renders u or any of its differential coefficients infinite.

(L. C. D. 169.)

Maclaurin's Theorem. Let (u)o, du represent the values of u, du, respectively, when x=0, then

(9.) Taylor's theorem applied to two variables.

Ex,(u)=u+ {du. Dx +du. Dy}

1

+ {d2u (Dx)2 +2d ̧du.Dx.Dy+du(Dy)2}

1.2

(10). Laplace's Theorem. Let y={≈+x.p(y)}, in which ≈ is independent of a and y, then

f(y)=ƒ¥(x)+= {¢¥(*).d2ƒ¥(*)}+~.d.{p¥(*)}*°•d_ƒ¥(x)}

3

1.2

В

+ ··d2 { p¥(≈) |3.d2ƒ¥ (≈)} + &c.
1.2.3
Lagrange's Theorem. Let y=x+x.(y), then

α

1

X3

+ •d2 {p(≈)]3·d_f(x)} + &c.

1.2.3

γ

This theorem is a particular case of the former, in which
¥ {≈+x.p(y)} =2+x.
c.p(y).

1

E,,(u) = SS, (Dx)m-n. (Dy)n-1.d ̧m-n.d ̧n-1:u.

m

If a third variable ≈ be supposed to receive an increment, then

Maclaurin's theorem applied to a function of two variables.

(Jarrett, Camb. Phil. Trans. Vol. III, Pt. 1.)

(L. C. D. 107—9; Tr. L. Note E; L. D. C. 49—61.)

IMPLICIT FUNCTIONS.

(11.) Let du=M, du= N, then

M+Nd,y=0,

from which equation the value of dy may be found. By successive differentiation, the following equations may be obtained, in which u1, u2, &c. are functions of x and y:

u1+uşd ̧y+u ̧(d ̧y)2 + Nd2y=0,

2

3

uş+uçd ̧y+uç(d ̧y)2 + u(d ̧y)3 + (uç+uçd ̧y)d2y+Nd3y=0,

from which the values of day, day, &c. may be obtained.

(L. C. D. 41-8; L. D. C. 100-3.)

If x and y are considered as functions of a third variable t,

day=

dx.d2y-d2x.dy
(d+x)3

(dx)2. d3y — 3 dx.d2x.d2y+3 (d2x)2.dy—dx.d3x.d ̧y

&c. &c.

∞

y = x + S

(13.) Elimination of an arbitrary function.

Let x= pf(x,y)=$(t) suppose, then

dx=d,p(t).d ̧t, and d ̧x=d.p(t).d1t, and hence

In the same manner, by repeating the differentiation of any given equation, two or more arbitrary functions may be (L. C. D. 77—9.)

eliminated.

PARTICULAR VALUES OF THE VARIABLE.

(14.) Let u=0, when x=a, then u must be of the form

v being a function of a not containing the factor — a.

If m=n, then dru=vn which is the first differential coefficient that does not vanish, when x = a.

If m is not an integer, then da+lu and all succeeding coefficients become infinite, when x=a.

Let n, r be the greatest integers m, p respectively, then when xa, if n>r, u=0;

n<r, u, and all its differential coefficients are

If u =

f(x)

0

=

p(x)

when a = a, the value of u may be thus

found: substitute a + Dx for x in f(x) and p(x), and expand in series of ascending powers of Dx, then

If u=f(x).f1(x)=0.∞, when x=a, put p(x)=

1

f(x)

under which form the value of u may be determined as above.

(L. C. D. 12; L. D. C. 107.)