that is to say, that in order to obtain the differential of the product of two functions of x, each must be multiplied by the differential of the other, and the two results added together. = If we have y = a. u, a being constant, and u a function of x, we have, according to the proposition proved above [since da O, (art. 8.)], dy Ξα. du. Next, taking three functions of x, let y = u . v . z; supposing u . v = n, we have dy =ndz + zdn, dn, = udv + vdu; substituting the values of n and of dn in the value of dy, we have (art. 10.) dy uzdv vzdu + uvdz. Hence results the mode of finding the differential of a product of the functions of the same independent variable, whatever their number may be; thus: multiply the differential of each of these functions by the product of all the other functions, and add together the results. 12. We are now able to differentiate the equation y=x". We have ☛" = x . in which the functions of a generally denoted in the preceding number by u, v, z . . . are consequently all equal to each other and at the same time equal to x. Each of the terms forming the differential will then be equal to x-1 dx. Now these terms are in number n; we have therefore dy = d. =nx2-1 dx. According to art. 9, we shall easily find the differential of y = x”, y being the independent variable, or of x = Thus, we have (art. 9.) Hence may be easily found, with the help of art. 10, the differential of Thus, in the function y = a", whether m be integral or fractional, but a positive number, we have always dy = d. xm = mxm-1dx. The rule is to find the differential of am, m being an integral or fractional positive number, multiply da by the exponent of the independent variable, and by the proposed power, its exponent being diminished by unity. Making the last fraction equal to the following series, in which a, ß, a Ax + BAx2 + . are indeterminate co-efficients, we obtain dv Ax+v"Ax 2 + dx Hence it results that the differential of a fraction whose numerator and denominator are functions of the same variable, is equal to the differential of the numerator multiplied by the denominator, minus the differential of the denominator multiplied by the numerator, and the result divided by the square of the denominator. Let there be given the equation whence we may infer that the differential of ", in which m is a negative number, integral or fractional, will be mxx-1dx. Comparing this with what was said in the preceding article on the differen tial of aTM (in which m was regarded only as a positive number) we obtain the following proposition: da" is equal to ma-ide, whether m designates an integral or fractional, positive or negative number; that is to say, the differential of aTM, whether m be a positive, negative, integral or fractional number, is equal to da multiplied by the exponent of the variable, and by the power in which the exponent m is diminished a unit. 14. The rules enunciated in the preceding pages will be sufficient for the differentiation of all the algebraic functions. Some examples are added, of the application of these rules. Ex. 1. Let y = (a + bxTM)". Putting a + bxm=z, we have y = 2", dy nz"-1dz (art. 12); therefore (art. 10), dy = nmb (a + bx”)-1, xTM-1 dx. Comparing this function with u and v in art. 13, we have u = ax, du a da (art. 11), The reader will do well to differentiate the following functions, carefully applying the enumerated rules, as has been done in the two preceding examples. dy da 15. The differential coefficient =p being a function of a (art. 6), we may obtain from it, as from a primitive function, the difference, the ratio of the differences, and the differential coefficient. Now, the notation of the differential dy dy coefficient is da if y is the primitive function; if then, we take as a pridx' mitive function, we should, from analogy, designate the differential coefficient of d (dy dy) day ; but it is usually designated by da2' or the differential coefficient by da by f" (x); and frequently by the letter q. It is properly called the second differential coefficient, or the differential coefficient of the second order of the function y. It is also written, as has been explained, for the first differential coefficient (art. 6), under this form, considering in this expression d'y as the second differential of y, and dæ2 the square of the differential of x. d2y Now being considered as the primitive function, we obtain the third dif ferential coefficient of y, that is to say, d2y da2 This notation is most fre dx d3y quently superseded by by r, or by f'(x). The third differential of the From this the notation of differential coefficients, and of the ulterior differentials, may easily be settled. Differential coefficients, and differentials above the first order, are called differential coefficients of the superior orders, or superior differential coefficients, and superior differentials of the primitive function y. It is easily seen that the second differential in such manner that dy and da are considered as true quantities, and pdæ as a product; dy and p as variables, but da as constant. In fact, we then obtain In the same way the third differential may be obtained from the second differential d'y = q. da2, differentiating so as to regard d'y and q as variables, and da2 as constant; and so on for successive differentials. Having in the preceding articles explained the method of differentiating any algebraic function, we are now prepared to find also the superior differentials, each of the successive differential coefficients of an algebraic function being also an algebraic function. We may develop, by way of example, the differentials of the superior orders of the function y = ax". We have (art. 12) Ex. 2. Find the successive differential coefficients of y = ax + bx3 + cx1 + dx3 + ex2 + gx + k. ON THE TRANSFORMATION OF A FUNCTION IN A SERIES ARRANGED ACCORDING TO THE INTEGRAL POWERS OF THE VARIABLE. 16. We shall now, by means of the superior differentials, express a function of a single variable by a series arranged according to the integral powers of this variable. |