The Calculus Primer (2011)

Part VII. Differentials

Chapter 25. INCREMENTS AND INFINITESIMALS

7—1.The Meaning of a Differential. Let us consider again the function which was employed in §2—4 and §2—5 to develop the meaning of a derivative, namely, y = x². We learned that as x took on an increment Δx,

Δy = 2x·Δx + (Δx)².(1)

We know that, for this function, = 2x; hence we may write equation (1) as follows, where y′ = :

Δy = y′Δx + (Δx)².(2)

We wish to study the expression y′Δx, and thus introduce these new terms.

(1) The differential of an independent variable is the same as its increment. The symbol for the differential of x is dx; hence dx = Δx.

(2) The differential of a function (or dependent variable) is the product of its derivative and the differential of the independent variable. The symbol for the differential of a function is dy; hence dy = y′dx, or y′ = .

It will be seen that equation (2) can also be written as

Δy = y′dx + (dx)²,

orΔy = dy + (dx)².(3)

Up to this point in our study of the calculus we have stressed the fact that was not to be regarded as the ratio of two quantities, but rather as a single symbol denoting the limiting value of , where Δy and Δx represented two separately changing quantities. We now see that the meaning of the symbol may be extended. The derivative may now be thought of as a quotient of two differentials, or

= dy ÷ dx.

7—2.Geometric Interpretation of dx and dy. This extension of meaning will become clearer from a study of the figure. For a given increment to x equal to Δx, the corresponding increment of y is Δy = RQ. The tangent PT,having an inclination ø, and a slope equal to = = tan ø, cuts off a part of Δy, namely, RS; this part of Δy equals dy. The segment MN = Δx, and PR = dx; thus Δx = dx.

We may regard dy as the value of Δy in the limit when the secant coincides with the tangent.

The relations between increments, differentials, and derivatives are extremely important and should be thoroughly understood. It should be noted that Δx and dx are quantities measured in the same unit as x, and Δy and dy are quantities measured in the same unit as y. The derivative, , that is the slope, when considered as a single quantity, or rate (not a ratio of two quantities), is generally measured in a compound unit such as miles per hour or feet per second.

7—3.Relation between Δy and dy. The precise nature of the differential dy will be better understood from the following considerations. Originally, we defined the derivative with respect to x of a function y = f(x) as the limit of the difference-quotient; that is

Since the value of at each stage of the process of passing to the limit, when x is fixed, depends upon the corresponding value of Δx, the difference-quotient is a function of Δx.

By the definition of the limit of a function, the difference between and f′(x) can be made as small as we please by taking Δx sufficiently small. This idea may be represented symbolically by

where € represents some variable quantity which is approaching zero in value. Equation (1) may be written as

Now the product of a small quantity and another small quantity yields a much smaller quantity; for example, To be sure, the terms “small” and “much smaller” are purely relative, and are admittedly used in a somewhat loose sense; but they may help us to understand the ideas involved. Thus, is, in a sense, of a different “degree of smallness” than or .

Let us turn to equation (2) once more. As Δx becomes smaller, the quantity € also becomes smaller; the product of € and Δx is of a lesser degree of smallness than Δx. Hence, in the right member of the equation the sum usually consists of a relatively large part, f·(x) · Δx, and a relatively small part, €·Δx. The larger part, f′(x)·Δx, may be called the principal part of Δy; and by disregarding the comparatively negligible part €·Δx, we see that Δy is approximately equal to its principal part, f′(x)·Δx. For example, if y = f(x) = 10x²,

Δy = f′(x)Δx + e· Δx,

then Δy = 10(x + Δx)² − 10x² = 20x·Δx + 10 (Δx)²;

also,f′(x) = 20x.

Hence,Δy = f′(x)Δx + 10Δx·Δx,

which yields € = 10Δx. If we arbitrarily take x = 1 and Δx = .01, we have

Δy = (20) (.01) + (10) (.01) (.01) = .2 + .001 = .201.

Thus the value of Δy, .201, is approximately equal to its principal part, f′(s)Δx = .200.

The differential of a dependent variable may be defined as the principal part of Δy; or

dy = f′(x)·Δx.(3)

The value of dx is always taken as equal to Δx, for any given function, and is regarded as a constant. The value of dy, however, depends upon the values of both x and Δx; for a fixed value of x, the value of dy varies directly as the value of Δx.

We may therefore now finally write:

dy = f′(x) dx.(4)

Equation (4) may be interpreted to mean that the derivative is considered as a ratio, or as the quotient of two differentials, for now becomes dy ÷ dx. In words, the differential of a function equals the product of its derivative and the differential of the independent variable.

From now on we may therefore regard the symbols dy and dx as separate quantities (differentials), and the symbol as a fraction.

7—4.Infinitesimals. We now see that an infinitesimal may be defined as a variable whose numerical value becomes and remains smaller than any preassigned value, however small. Or, an infinitesimal is a variable which approaches zero as a limit.

When comparing infinitesimals, we refer to their order. This is a relative term, suggesting comparative degree of smallness. If the limit of the quotient of two infinitesimals is a constant, not zero, they are said to be of the same order; if this limit is zero, the first differential (the numerator) is said to be of higher order than the second, and the second of lower order than the first. If the limit is infinite, the first differential (the numerator) is said to be of lower order than the second, and the second of higher order than the first.

EXAMPLE 1.In §7—3, when discussing the relation

Δy = 20x·dx + 10(dx)²,

we saw that, as dx → 0, the quantity 10 (dx)² approached zero “faster” than did the quantity 20x·dx; in other words, 10(dx)² is of a higher order than 20x·dx because

EXAMPLE 2.In , the two infinitesimals sin θ and tan θ are of the same order, since

Thus, in general,

where f′(x) ≠ 0; therefore Δy and dy are infinitesimals of the same order. On the other hand, from the relation

Δy − dy = €·Δx,

it will be seen not only that the infinitesimal €· Δx is of a higher order than either Δy or dy, but also that dy is an approximation to Δy; the smaller the value of Δx, the more closely dy approximates Δy.