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 ƒ ... dx represents the inverse operation. If the former is denoted by D, the latter may be denoted by D−1. Dn means that Symbols of operation. the operation D is to be performed n times in succession; D−n that the operation of forming the indefinite integral 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 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: 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
| ƒ(b) − ƒ(a) | = ƒ′(x1). |
| b − a |
(i.)
The value x1 can be expressed in the form a + θ(b − a) where θ is a number between 0 and 1.