Page:EB1911 - Volume 14.djvu/575

finding out whether this condition is satisfied or not is to attempt to form the differential coefficient. If the quotient of differences Δy/Δx has a limit when Δx tends to zero, y is a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function ƒ(x) is always defined by the formula

$f^{\prime }(x)={\frac {df(x)}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$.

Rules for the formation of differential coefficients in particular cases have been given in §11 above. The definition of a differential coefficient, and the rules of differentiation are quite independent of any geometrical interpretation, such as that concerning tangents to a curve, and the tangent to a curve is properly defined by means of the differential coefficient of a function, not the differential coefficient by means of the tangent.

It may happen that the limit employed in defining the differential coefficient has one value when h approaches zero through positive values, and a different value when h approaches zero through negative values. The two limits are then called the “progressive” and “regressive” differential coefficients. Progressive and Regressive Differ-ential Coefficients. In applications to dynamics, when x denotes a coordinate and t the time, dx/dt denotes a velocity. If the velocity is changed suddenly the progressive differential coefficient measures the velocity just after the change, and the regressive differential coefficient measures the velocity just before the change. Variable velocities are properly defined by means of differential coefficients.

All geometrical limits may be specified in terms similar to those employed in specifying the tangent to a curve; in difficult cases they must be so specified. Geometrical intuition may fail to answer the question of the existence or non-existence of the appropriate limits. In the last resort the definitions of many Areas. quantities of geometrical import must be analytical, not geometrical. As illustrations of this statement we may take the definitions of the areas and lengths of curves. We may not assume that every curve has an area or a length. To find out whether a curve has an area or not, we must ascertain whether the limit expressed by ∫ydx exists. When the limit exists the curve has an area. The definition of the integral is quite independent of any geometrical interpretation. The length of a curve again is defined by means of a limiting process. Let P, Q be two points of a curve, and R₁, R₂, . . . R_n−1 a set of intermediate points of the curve, supposed to be described in the sense in which Q comes after P. The points R are supposed to be reached successively in the order of the suffixes when the curve is described in this sense. We form a sum of lengths of chords

PR₁ + R₁R₂ + . . . + R_n−1Q.

If this sum has a limit when the number of the points R is increased indefinitely and the lengths of all the chords are diminished indefinitely, this limit is the length of the arc PQ. The limit is the same whatever law may be adopted for inserting the intermediate points R and diminishing the lengthsLengths of Curves. of the chords. It appears from this statement that the differential element of the arc of a curve is the length of the chord joining two neighbouring points. In accordance with the fundamental artifice for forming differentials (§§ 9, 10), the differential element of arc ds may be expressed by the formula

$ds={\sqrt {(dx)^{2}+(dy)^{2}}}\,$,

of which the right-hand member is really the measure of the distance between two neighbouring points on the tangent. The square root must be taken to be positive. We may describe this differential element as being so much of the actual arc between two neighbouring points as need be retained for the purpose of forming the integral expression for an arc. This is a description, not a definition, because the length of the short arc itself is only definable by means of the integral expression. Similar considerations to those used in defining the areas of plane figures and the lengths of plane curves are applicable to the formation of expressions for differential elements of volume or of the areas of curved surfaces.

34. In regard to differential coefficients it is an important theorem that, if the derived function ƒ′(x) vanishes at all points of an interval, the function ƒ(x) is constant in the interval. It follows that, if two functions have the same derived function they can only differ by a constant. Conversely, indefinite integrals are indeterminate to the Constants of Integration. extent of an additive constant.

35. The differential coefficient dy/dx, or the derived function ƒ′(x), is itself a function of x, and its differential coefficient is denoted by ƒ″(x) or d²y/dx². In the second of these notations d/dx is regarded as the symbol of an operation, that of differentiation with respect to x, and the index 2 means Higher Differential Coefficients. that the operation is repeated. In like manner we may express the results of n successive differentiations by ƒ⁽ⁿ⁾(x) or by dⁿy/dxⁿ. When the second differential coefficient exists, or the first is differentiable, we have the relation

$f^{\prime \prime }(x)=lim_{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}$

(i.)

The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′(x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ(x) must be a linear function of x in the interval, is important.

The relation (i.) is a particular case of the more general relation

$f^{(n)}(x)=\lim _{h\to 0}h^{-n}{\bigg [}f(x+nh)-nf\left\{x+(n-1)h\right\}+{\frac {n(n-1)}{2!}}f\left\{x+(n-2)h\right\}-\ldots +(-1)^{n}f(x){\bigg ]}$

(ii.)

As in the case of relation (i.) the limit expressed by the right-hand member may exist although some or all of the derived functions ƒ′(x), ƒ″(x), . . . ƒ⁽ⁿ⁻¹⁾(x) do not exist.

Corresponding to the rule iii. of § 11 we have the rule for forming the nth differential coefficient of a product in the form

${\frac {d^{n}(uv)}{dx^{n}}}={\frac {d^{n}v}{dx^{n}}}+n{\frac {du}{dx}}{\frac {d^{n-1}v}{dx^{n-1}}}+{\frac {n(n-1)}{1\cdot 2}}{\frac {d^{2}u}{dx^{2}}}{\frac {d^{n-2}v}{dx^{n-2}}}+\ldots +{\frac {d^{n}u}{dx^{n}}}v$,

where the coefficients are those of the expansion of (1 + x)ⁿ in powers of x (n being a positive integer). The rule is due to Leibnitz, (1695).

Differentials of higher orders may be introduced in the same way as the differential of the first order. In general when y = ƒ(x), the nth differential dⁿy is defined by the equation

dⁿy = ƒ⁽ⁿ⁾(x)(dx)ⁿ,

in which dx is the (arbitrary) differential of x.

When d/dx is regarded as a single symbol of operation the symbol ∫. . .dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D⁻¹. Dⁿ means that the operation D is to be performed n times in succession; D⁻ⁿ that the operation of forming the indefinite integralSymbols of operation. is to be performed n times in succession. Leibnitz’s course of thought (§ 24) naturally led him to inquire after an interpretation of Dⁿ where n is not an integer. For an account of the researches to which this inquiry gave rise, reference may be made to the article by A. Voss in Ency. d. math. Wiss. Bd. ii. A, 2 (Leipzig, 1889). The matter is referred to as “fractional” or “generalized” differentiation.

Fig. 9.

36. After the formation of differential coefficients the most important theorem of the differential calculus is the theorem of intermediate value (“theorem of mean value,” “theorem of finite increments,” “Rolle’s theorem,” are other names for it). This theorem may be explained as follows: Theorem of Intermediate Value. Let A, B be two points of a curve y = ƒ(x) (fig. 9). Then there is a point P between A and B at which the tangent is parallel to the secant AB. This theorem is expressed analytically in the statement that if ƒ′(x) is continuous between a and b, there is a value x₁ of x between a and b which has the property expressed by the equation

${\frac {f(b)-f(a)}{b-a}}=f^{\prime }(x_{1})$.

(i.)

The value x₁ can be expressed in the form a + θ(b − a) where θ is a number between 0 and 1.

A slightly more general theorem was given by Cauchy (1823) to the effect that, if ƒ′(x) and F′(x) are continuous between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

${\frac {F(b)-F(a)}{f(a)-f(b)}}={\frac {F^{\prime }\left\{a+\theta (b-a)\right\}}{f^{\prime }\left\{a+\theta (b-a)\right\}}}$.

The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where ƒ(x) is a rational integral function which vanishes when x = a and also when x = b. The general theorem was given by Lagrange (1797). Its fundamental importance was first recognized by Cauchy (1823). It may be observed here that the theorem of integral calculus expressed by the equation

$F(b)-F(a)=\int _{a}^{b}F^{\prime }(x)dx\,$

follows at once from the definition of an integral and the theorem of intermediate value.

The theorem of intermediate value may be generalized in the statement that, if ƒ(x) and all its differential coefficients up to the nth inclusive are continuous in the interval between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation

$f(b)=f(a)+(b-a)f^{\prime }(a)+{\frac {(b-a)^{2}}{2!}}f^{\prime \prime }(a)+\ldots$

$+{\frac {(b-a)^{n-1}}{(n-1)!}}f^{(n-1)}(a)+{\frac {(b-a)^{n}}{n!}}f^{(n)}(a)\left\{a+\theta (b-a)\right\}$.

(ii.)

37. This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated series, and, when the remainder tends to zero as n Taylor’s Theorem. increases, it may be transformed into an infinite series. The theorem