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 essen tial 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: x", m being any real quantity whatever, a, sin x, cos x, tan x, cot x, sec x, cosec x, and the inverse functions log. x, sin1x, cos1 x, tan1 x, cotx, secx, cosec1 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 x" with respect to x, n being any rational quantity whatever. 30. Put y = x"; then, x', y', being corresponding values of x and y, we have 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 1 Hence, dividing vo – 1, va − 1, v′′ – 1, by v 1, observing that p, q, r, are positive integers, we have Now, by making 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 expression n 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 da 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 log. (1+1), 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 (v + 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 a with regard to x. Now, to proceed to the limit, putting n = an indefinitely large positive integer, and thereby rendering dx less than any assignable quantity, we have, To find the Differential Coefficient of sin x with regard to x. Now by the seventh Lemma of the first section of Newton's Principia we know that the arc and the chord of any curve vanish in a ratio of equality: whence it follows that the ratio between the sine and the circular measure of an angle is ultimately unity. Hence, in the limit, 34. Put then dy dx = COS X, dy = cos x . dx. To find the Differential Coefficient of cos x. |