whence, proceeding to the limit, that is, equating de to zero, and therefore, y, and y, being supposed to be continuous functions of x, putting also dy, and dy, each equal to zero, we have 2 du = y1 dy2+ y, dy,. Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied by the differential coefficient of the other. Taking x', u', y, y', to denote simultaneous values of х, и, У1, У2, we have and therefore, proceeding to the limit, that is, equating x-x Ꮖ or da to zero, we get, observing that y, becomes y2, Hence, to differentiate the Ratio of two functions, we have the following rule: Multiply the denominator by the differential coefficient of the numerator, and the numerator by the differential coefficient of the denominator: subtract the latter product from the former: this difference divided by the square of the denominator is the differential coefficient of the Ratio. Differentiation of the Product of any number of Functions. 17. If u y1.y2.y.... y, the product of n functions of x, then = 2 3 Since u uy, we have, by Art. (15), whence 92 = similarly и n-1 =3 dx u 12-2 + dx n-1 n-2 Yn-1 dy-1 dx adding these equations together, cancelling terms which are common to both sides of the resulting equation, and observing Relation between Inverse Differential Coefficients. 18. If y be a function of x, in which case x will also be a function of y, then Let x', y', be simultaneous values of x, y; then it is evident and that, consequently, proceeding to the limit, Differentiation of a Function of a Function. 19. If u be a function of y, and y a function of x, then COR. If u be a function of y,, y, of y2, y2 of y.,....and y, of x, it is manifest that we may prove in the same way that Differentiation of a Function of two Functions. 20. Let u = f(y1, y), where f(y,, y) denotes any function whatever of y, and y2, each of the quantities y, and y, being a function of a third quantity x. Let y1, y2, u, become y', y', u', when x becomes x': then u = f(y1, Y2), u' = f(y1', Y2), u' – u = f(y1', y2) − f(Y1, Y2) 12-Y2 x'- X Now in the limit, when x' differs from x less than by any assignable magnitude, and, first replacing y, by y1, and then y, by y2, dy, dx = df (y1, y2) du dy, dy2 2 ; In this equation it is very important to observe that the numerators of the two fractions although represented by the same symbol du, are essentially different, the numerator of the former corresponding to the ultimate value of the increment f(y, y2) − f(y1, Y2), and the numerator of the latter to the ultimate value of the increment and which we have represented by a distinct symbol Du, is evidently different from either of the numerators of the fractions du du dy' dy In order to obviate all possibility of confusion, we might use du du dy,' dy, the symbols du, du, to denote the numerators of the suffixes serving to point out the origin of the two differentials. Such a notation, however, although remarkably clear, would frequently be very embarrassing, especially in long operations. It will be sufficient for distinctness if we remember to regard du dy, du and as fractions the denominators of which are insepa dy2 rably attached to the numerators, the symbols dy, and dy,, which express the denominators, thus serving to indicate the true nature of the du in the numerators. If, however, as will be sometimes convenient, we do put du, du du du, instead of du, in the expressions dy,' dy, , we shall then be at liberty to treat these differential coefficients as ordinary algebraical fractions: thus may be written, multiplying both sides of the equation by dx, Y 2 The quantity du denotes the differential of u taken with regard to Y1 as if y, were constant, du the differential of u taken with regard to y, as if y, were constant, and Du the differential of u due to the simultaneous variations of y, and y, dependent upon the variation of x. The quantities du, du, are called the partial differentials of u with regard to y1, y2, respectively, and Du its total differential. The quantities |