In this case d'y = f'(x) dx = ƒ'(x) eo də, d"3y = ƒ"(x) dx eo d0 +ƒ'(x) eo d02 = ƒ ̈'(x) e2o d02 + ƒ'(x) e" d03, d*y = f(x) dc e đôi +2f (2) e** dô* + f (x)dx e® đô + f(x) e® ® = ƒ"(x) e3o d03 + 3ƒ ̈(x) e2o dμ3 + ƒ'(x) eo d03, Order of Partial Differentiations indifferent. 48. The following is a theorem of great importance in successive differentiation: if In precisely the same way it may be shewn that §ï ̧§Ã ̧u=ƒ(Y2+ dy ̧‚ Y2+ dy2} − ƒ(y,+dÿ ̧‚ Y2)−ƒ(Y1, Y2+ dy2)+f(Y1, Y2)• The right-hand members of these last two equations being identical, we must have also This relation is true whatever be the magnitudes of Sy, and dy,: if we proceed to the limit, by taking dy, and dy, less than any assignable magnitudes, and replace infinitesimal differences by differentials, we get Expressing the theorem by partial differential coefficients instead of partial differentials, we have or, as these partial differential coefficients are ordinarily expressed for the sake of brevity, it is evident that the symbols d,, dy, of partial differentiation, may be permuted in every possible way: thus dy dru = dv, dv, dvu = dy, dvdyu = dv, dy, dvu = dv, vu, 2 or, in the language of partial differential coefficients, COR. 2. The theorem which we have established in relation to partial differentiation of functions of two variables, may evidently be extended to the general case of a function of any number of variables: thus, in an expression u being a function of y,, y2, Y,,... the symbols dy, dy, dy,.... may be permuted inter se in the same way as the symbols of quantity, A ̧, ^,  ̧‚. ...in an algebraical product 39° Successive Differentiation of an explicit Function of two 49. Let u = f(y1, y2), y, and y, being each of them a function of x then, by Art. (20), Differentiating again, x being considered the independent variable, and observing that, for convenience of writing, we may put, V being any expression functional of x, = dx2 dx dy d dy_dy2: = dx dx dx dx dx2 dy, D (du dy, du dy du d'y2 2 + + + dx dx dy2. dx dy dx dy, dx2 Now is equivalent to a function of y, and y, only, not du dy, involving the differential dy,: thus, for instance, if u = 2y,y,, where dy, does not appear. It follows there fore that in the expression Ddu dx dy, we may regard dy, constant without affecting results. Hence, du dy, now occupying the place of u in (1), this formula gives We might proceed in the same way to find the expressions for D3u, D3u,......; the formulæ however rapidly rise into tedious polynomials. We have confined our attention to the successive differentiation of a function of two functions; the extension however of the theory to a function of any number of functions is too obvious to present any difficulty to the student. Thus, supposing that |
