not take any arbitrary value. Thus, in the equation ay + bx = xy, making x equal to c, we have y =

bc

c— a

In this example, after determining the arbitrary value of x, we have found the value of y. But we might otherwise begin by determining the arbitrary value and then finding x in the equation.

of

y,

In an equation of two variables, that which may be determined at pleasure is called the independent or primitive variable, and the other, whose value must afterwards be found in the given equation, is called the dependent variable.

We shall for the future represent the independent variable by the letter x, and the dependent variable by the letter y, except where a particular limitation is made.

In an explicit equation of two variables, the dependent variable y as a function is sometimes expressed by f (x) or ø (x).

The form, or general expression of explicit equations of two variables, is y = f(x), when ƒ (x) contains only the single independent variable x, besides the constant quantities.

5. Let us now place, in an explicit equation of two variables, x + Ax instead of the independent a; the function y = ƒ (x) is then changed into

y + Ay = f (x + Ax).

According to this notation, Ax indicates the change or difference of the independenta, Ay the change or difference of the dependent or function y, which results from it; then a + Ax will indicate the modified independent variable, and y + Ay the modified dependent variable. We express by ƒ (a) the proposed or primitive function, by y = f(x) the proposed or primitive equation. We now propose to develope ƒ (x + ▲) in a series arranged according to the integral and positive powers of Ax. Let

A, B, C, &c. contain only a, and not ▲x; we must now determine them. The coefficient A may be immediately determined; for when we make in the above series Ax = O, Ay will also be equal to zero, and the series is reduced to its first term; we then have y A, that is to say, A is the primitive function. Removing y = A from the equation, we have

Ay = B. Ax + C . Ax2 +

or the change of a function expressed by a series arranged according to the integral powers of Ax.

Example. Let the primitive function be y = ax2; we then have

y + ▲y = a . (x + ▲x)2 = ax2 + 2ax▲x + a▲x2;

taking away y =ax2, there remains

Ay 2αxAx + aax2.

So that the coefficients A, B, C, D .. are here determined by the values ax2, 2ax, α, 0...

6. Dividing by Ar the two members of the series,

= y' + y'Ax +

where y'y"Ax + is the exponent of the ratio of the two changes, Ax and Ay, or the ratio of the differences. The first term of the ratio of the differences is therefore y'. The notation most in use for this term, besides f'(x), of which we The first notation of the first term of Ay indicates

Δα

dy have treated above, is da that it is a function of x, and more particularly a function derived from y = f(x) (art. 5); the last announces that it has its origin in monly called the differential coefficient, which it is the object of the differential calculus to find. The expression y':

dy

dy

Ay dy
is com-
Ax da

dy dx

is usually denoted by dy = y'da, or

= da, and dy is called the differential of y, dx that of x; and the dif

da

ferential of y is said to be equal to the quantity y' or dy, multiplied by the

dx'

differential of x. But dy and dæ must not be regarded as true quantities in

dy
dx

dy

themselves, nor y'da and da as real products, nor as a real quotient.

dx

When henceforward we seek the differential of a function, we shall only require to find the differential coefficient of that function, and then express it under the form of the differential.

dy
dx

The differential coefficient is also expressed, often advantageously, by p, and consequently the differential of a function y by dy = p. dx.

We have found in the above example

We have just shown that if y is the function of x, we have

Consequently, we have in the primitive equation x = ƒ (y), considering a reciprocally as the function of y,

Moreover, if for example z is the function of u, we have

sented by du, which may be substituted for it.

Finally, we may derive the difference from a given differential; suppose

Ay

it will be at once evident that the value of (that is to say, the series

Δη

dy

dx

+ y′′Ax + .), supposing Ax = 0, reduces itself to

dy

the differential coefficient. This value will differ very slightly indeed from dx if Ax is supposed very small; and this difference will continue to diminish as Ax becomes smaller. We may therefore regard as the limit of the value of da

dy

Ay since approaches this limit in proportion as Ax becomes smaller, but that Δη

Ay

Δι

it only reaches the limit when Ax is equal to zero.

dy
is therefore the limit of
dx

the ratio of the differences of y and x; which is also called the last or ultimate ratio of the differences of x and y themselves. According to this, it will

dy Ay
dx

be seen (art. 6.) how the notations and express the intimate connection

Ax

which subsists between the values which they indicate.

8. It will also be easily seen, after what has been said, how to proceed with regard to functions which contain many terms. Let

Putting + Ax instead of x, we have (according to art. 5 and 6), instead of u,

that is to say, that a function composed of many functions of the same independent variable, joined by means of addition or subtraction, has for its difference the sum of the differences of those functions, and for its differential co-efficient the sum of the differential co-efficients of each of those functions.

dy = du + dv + dz +

a proposition which is thus expressed; the differential of a function which is composed of several functions of the same independent variable, is equal to the sum of the differentials of each of those functions.

Thus, for example, we have in the function y = ax2 + bx,

dy =2axdx + bdx.

When one of the terms of which y is composed is constant, it is manifest that this portion has no influence in the differential of the function. Thus the functions

y = ax2 + bx and y = ax2 + bx + c,

have the same differential.

9. In the preceding article we have considered x as the independent in the equation y = f(x). Now let y be the independent. According to the first supposition, we have (art. 6,)

Ay

Ax

=

dy
dx

+ y" ▲ x +

member of this equation, y being now the independent

Ay + x" Ay2 +

Placing in the second (according to art. 6,)

dy

In determining a by the method of indeterminate coefficients, vol. i. p. 267, we obtain

an expression of the differential coefficient of the function a =

y is considered as the independent.

in which

a

dy
da

precisely as the frac

Hence it is evident that we may work the expression tion of which dy is the numerator, and do the denominator; notwithstanding which, the assertion made in art. 6, that it is not a true fraction, remains in force.

10. Let y be expressed by z, and z by x, so that y shall be a mediate function of the independent x, as in the equation y = z2, in which z is supposed = ax. According to this supposition, we have

that is to say, that in order to obtain the differential coefficient of a function which is mediate of a and immediate of z, the two differential coefficients of the mediate function (y) and the immediate (2) must be multiplied.

We may also observe, that the same result may be obtained by substituting in the expressions

the value of dz, obtained from the last of these expressions, in the first; we then have, as above,

Example. Required, to differentiate y = 22, when zax.

a result which would have been immediately obtained by differentiating the equation y

= a2 x2.

11. Suppose y = u v; u, v being functions of x. Substituting x + Ax for x, we obtain

y + Ay = (u + Au). (v + Av) = uv + vau + uav + Au. Av. Taking away the primitive equation, we have

Ay vAu + UAV + Au. Av.