may be regarded as a function of r independent variables X1, X2, X3,.•. .X ̧• We are therefore at liberty to consider X2, X.,....x,, as all constant, and to regard y1, y2, Y3. .. .Yn, as functions of a variable x,. We have therefore, by Art. (23), Similarly, ≈ ̧, ¤ ̧‚. .. .~,, successively taking the place of x1, + dy, dx, 0: = + = dx, + dx, dy, dx, dy2 dx, dy dx, There will evidently be also r analogous equations in relation to each of the functions v2, v39 V49 • • • • Vn° We thus have nr differential equations, and may thence determine expressions for the nr partial differential coefficients of the dependent variables y1, y2, Yз,.... Y viz. 22 „' in terms of the n (n+r) partial differential coefficients of v1, v2, v3, ... v, taken with regard to the variables x1, X2, X3, ... X,, SECTION II. SIMPLE FUNCTIONS. 29. In the preceding section we have shewn how to reduce the differentiation of a function of functions to that of the differentiation of its subordinate functions. In this section we shall investigate the differentials of what may be called simple functions, as being the constituent elements or subordinate functions of all the complex functions of algebra. The essential characteristic of a simple function consists in its not being susceptible of resolution into elements more simple than itself, except by the aid of infinite series: the number of simple functions might therefore, as may easily be imagined, be multiplied indefinitely. The algebraical expressions ordinarily adopted as simple functions are the following: xm, m being any real quantity whatever, a*, sin x, cos x, tan x, cot x, sec x, cosec x, and the inverse functions -1 log, x, sin ̈1x, cos1x, tan ̄1x, cot ̄1x, sec ̄1 x, cosec ̈1 x. These expressions have been selected as the elementary functions of ordinary analysis, in consequence of their peculiar utility in the various applications of the science. To find the Differential Coefficient of x1 with respect to x, n being any rational quantity whatever. 30. Put y=x"; then, x', y', being corresponding values of x y, we have and z being such a quantity that x' = xz. Our object is now to find the limiting value of the fraction when z approaches indefinitely near to unity. Now whatever be the value of n, positive, integral, fractional, or negative, we may always express it under the form where p, q, r, are positive integers. Hence Hence, dividing vo 1, va - 1, v" - 1, by v - 1, observing that P, q, r, are positive integers, we have p 1 (1 + v + v2 + • • • • + v31) − (1 + v + v2 +• . . . + v8-1). . . − +. Now, by making v approach more nearly to unity than by any assignable difference, z will also be made to do so: hence To find the Differential Coefficient of loga x with regard to x. Our object is to determine the value assumed by the expres when n = ∞, a value of n consequent upon the evanescent state of Sx. Now whether n be a continuous or a discontinuous variable, yet, provided that it become greater than any assignable magnitude, dx will become less than any assignable magnitude, which is the only condition to be fulfilled by dx in the ultimate state of the hypothesis. We will assume then n to represent a positive integer, and proceed to ascertain the limiting value of the function when the integer n becomes great without limit. By the binomial theorem we have the following expansion, Now we may take n and v so large that, if we stop the series at the (+1)th term, the sum of all the remaining terms will be less than any assignable magnitude. In fact, this sum is less than and becomes therefore indefinitely small when v is increased without limit. If then we take v indefinitely large, and neglect accordingly all terms after the (v + 1)th, and if we then take n, which is of course always larger than v, an indefinitely large number of a higher order of magnitude than v, so that in fact the ratio of n to v shall be indefinitely great, we shall have an approximation true without limit as v increases without limit; To find the Differential Coefficient of ax with regard to x. |