The quantity x, which is supposed to increase by equal augments, is called the independent variable, while y, the increments of which are dependent upon those of x, and which are generally variable, is called the dependent variable. Such is the definition of an independent and a dependent variable in the calculus of finite differences. Suppose now the difference dr to be indefinitely diminished, then we may replace

by the differentials

Sx, Sy, S3y, S3y,. . . . .

dx, dy, day, d3y,

....

which are proportional to them. We may then say, to adapt our definitions to the differential calculus, that if y be a function. of x, x will be the independent and y the dependent variable, if, while x varies, its differential de remains constant: in accordance with this definition not only y but also dy will generally vary with the variation of x.

=

Ex. 1. Let y sin x; then, ≈ being the independent variable, dy = cos x. dx:

differentiating again, ≈ and dy being variable, and de constant, d3y = d(cos x). dx

where da3, for simplicity of writing, is put instead of (dx)2. Proceeding in the same way, we see that

a rational function of x of n dimensions: then, differentiating successively n times, we have, x being the independent variable,

d2y dx2

d3y dx3

1.2.α, + 2.3.α2.x +. . . . . . + (n − 1) nax”-2,

dry

dx"

1.2.3.

....

n.a..

The differential coefficients of higher orders than the nth, viz.

Ex. 3. To find the nth differential coefficient of

2a

{(x − a) ̄1 − (x + a) ̃1}.

dry (1)"

dx"

.1.2.3. . . . . .n. {(x − a) ̄"-1 − ( x + a) ̄"-1}.

2a

Change of the Independent Variable.

47. Suppose that we have an equation

involving x, y, and successive differentials of y taken on the hypothesis that de is constant. It is frequently desirable in researches in the differential calculus to transform this differential equation into an equivalent one in which, instead of x, some quantity of which x is a function, shall be the independent variable. On the new hypothesis de will no longer generally be constant.

f'(x) being another function of x: adopting the same notation, put

df'(x) = f'(x),

dx

=

and so on. The quantities f'(x), ƒ (x), ƒ '(x),. .....are called the first, second, third,....derived functions or derivatives of f(x), and are certain algebraical expressions constituting the results of the operations upon the function f(x) designated by the differential coefficients

df(x) d2f(x) d3f(x)

Then, taking dy, d'y, d3y,.... to represent the differentials of y on the hypothesis that dx is constant, and d'y, d'3y, d'3y,. . . . its differentials, supposing de to vary, we have

d3y = ƒ"(x) dx3

d3y = ƒ" (x) dx" + 3ƒ (x) dxd3x + f(x) dox} · · · ·(3):

and so on to any order of differentiation.

Now, by the aid of the relation subsisting between x and 0, dx, d3x, d3x,. . . . may be found in terms of and do, and therefore, from (1), (2), (3),.... we can obtain dy, d3y, d3y,.... in terms of d'y, d'y, d'3y...., 0, do.

from (1), (2), (3), we may get also

dx (dxd"3y - d3xd'y) – 3d2x (dxd'y – d2xd'y)

d3y

dx2

(6):

The second equations of the systems (1), (2), (3),......may be written in a form which may serve to suggest to the memory that is the independent variable corresponding to the differentials d'y, d'y, d'3y,... ... of y, viz.

that is, the values of f'(x), f(x), ƒ"(x),. . . .

obtained from the equations (7), (8), (9),.... we shall have transformed the equation into an equivalent one

COR. Suppose that our object is to change the independent variable from x to y: then, by the formulæ (4), (5), (6), considering d'y constant, and therefore equating day, d'3⁄43y,....to zero, we have

to change the independent variable from x to 0, when x cos 0.

In this case d'y = f'(x) dx = -f'(x) sin 0 d0; differentiating again, considering de constant, we have

d'y = − ƒ"(x) dx sin 0 d0 – ƒ'(x) cos 0 d02