and (1 + a) is < 1+ ma (1 + a)”−1. From (2) it follows, a fortiori, that m-1 (2). Raise both sides of (5) to the nth power, and we have where μ is any positive fraction, and in (10) μẞ is < 1. Multiply both sides of (9) by 1+8; thus (1 + B)μ+1 > 1+ (μ + 1) ß, or, putting λ for μ + 1, (1 + ß)` > 1 + λß .... (11). Thus the inequality in (1), which required m to be a positive integer, is here extended, since λ may be any fraction or integer, provided it be greater than unity. From (12) we see that (1+) continually increases as x increases. It does not, however, pass beyond a certain finite limit; for in (10) write for B, and raise both sides 1 μη Hence, if we put y=2, we find that (1+ x 1 can never exceed 4. By ascribing to y greater values, we shall obtain a closer limit for (1+1)*. If y = 6, we see that (1+1)* where n has any positive value, we may, by ascribing successive integral values to n, easily approximate to the numerical value of the limit. CHAPTER II. DEFINITION OF A DIFFERENTIAL COEFFICIENT. DIFFERENTIAL COEFFICIENT OF A SUM, PRODUCT, AND QUOTIENT. 24. WE shall now lay down the fundamental definition of the Differential Calculus, and deduce from it various inferences. DEF. Let (x) denote any function of x, and 4 (x+h) the same function of x+h; then the limiting value of ☀ (x + h) − Þ (x), when h is made indefinitely small, is called h the differential coefficient of p (x) with respect to x. This definition assumes that the above fraction really has a limit. Strictly speaking, we should use an enunciation of (x+ h) - (x) this form-" If have a limit when h is made h indefinitely small, that limit is called the differential coefficient of (x) with respect to x." We shall shew, however, that the limit does exist in functions of every kind, by examining them in detail in this and the following two Chapters. We give two examples for the purpose of illustrating the defini tion. and the limit of 2x + h when h = 0, is 2x; therefore 2x is the differential coefficient of x2 with respect to x. a which is therefore the differential coefficient of +x with respect to x. b+ 25. We now give the notation which usually accompanies the definition in Art. 24. Let (x)=y, then (x+ h) - (x) is the difference of the two values of the dependent variable y corresponding to the two values, x and x+h, of the independent variable. This difference may be conveniently denoted by the symbol Ay, where A may be taken as an abbreviation of the word difference. We have thus Ay=4(x+ h) - $ (x). Agreeably with this notation, h may be denoted by Ax, so that ▲y_(x + h) - $ (x) Ay Ax = h It may appear a superfluity of notation to use both h and Ax to denote the same thing, but in finding the limit of the right-hand side we shall sometimes have to perform several analytical transformations, and thus a single letter is more convenient. On the left-hand side Ax is recommended by considerations of symmetry. We say then, according to the definition in Art. 24, that Ay the limit of when Ax is diminished indefinitely, is the (x) with respect to x. This differential coefficient of y or limit is denoted by the symbol dy T. D. C. C |