I) x ; a) (x + h) = x + 3hx + 3h x + h ; b) (x + h) - x = 3x h + 3xh + h ; c) ((x + h) - x )/h = 3x + 3xh + h . If h becomes = 0, then and the right-hand side = 3x , so that dy/dx = 3x . y = x ; y1 = x1 ; y1 - y = x1 - x = (x1 - x) (x1 + x1x + x ) ; y1 - y)/(x1 - x) or = dy/dx = x + xx + x ; dy/dx = 3x . ___________ II) Let us set x1 - x = h. Then: 1) (x1 - x) (x1 + x1x + x ) = h(x1 + x1x + x ) ; (y1 - y)/h = x1 + x1x + x . In 1) the coefficient of h is not the completed derivative, like f above, but rather f ; the division of both sides by h, therefore, also leads not to dy/dx, but rather y/h or y/ x = x1 + x1x + x If we begin on the other side in I c), namely in (f(x + h) - f(x))/h or (y1 - y)/h = 3x + 3xh + h , 3x thus becomes a value which the series constantly approaches, without ever reaching it, and thus, even more, without ever being able to exceed it. In this sense 3x becomes the limit value89 of the series 3x + 3xh + h . On the other side the quantity (y1 - y)/h (or (y1 - y)/(x1 - x)) also decreases all the more, the more its denominator h decreases.90 Since, however, (y1 - y)/h is the equivalent of 3x + 3xh + h the limit value of the series is also the ratio s own limit value in the same sens that it is the limit value of the equivalents series. However, as soon as we set h = 0, the terms on the right-hand side vanish, making 3x the limit of its value; now 3x is the first derivative of x and so = f (x). As f (x) it indicates that an f (x) is also derivable from it (in the given case it = 6x) etc., and thus that the increment f (x) or 3x is not = the sum of the increments which can be developed from f(x) = x . Were f(x) itself an infinite series, so naturally the series of increments which can be developed from it would be infinite as well. In this sense, however, the developed series of increments becomes, as soon as I break it off, the limit value of the development, where limit value here is in the usual algebraic or arithmetic meaning, just as the developed part of an endless decimal fraction becomes the limit of its possible development, a limit which is satisfactory on practical or theoretical grounds. This has absolutely nothing in common with the limit value in the first sense. Here in the second sense the limit value may be arbitrarily increased, while there it may be only decreased. Furthermore (y1 - y)/h = (y1 - y)/(x1 - x) , so long as h is only decreased, only approaches the expression 0/0; this is a limit which it may never attain and still less exceed, and thus far 0/0 may be considered its limit value.91 As soon, however, as (y1 - y)/h is transformed to 0/0 = dy/dx, the latter has ceased to be the limit value of (y1 - y)/h, since the latter has itself disappeared into its limit.92 With respect to its earlier form, (y1 - y)/h or (y1 - y)/(x1 - x), we may only say that 0/0 is its absolute minimal expression which, treated in isolation, is no expression of value (Wertausdruck); but 0/0 (or dy/dx) now has 3x opposite it as its real equivalent, that is f (x). And so in the equation 0/0 ( or dy/dx) = f (x) The concept of the limit value may therefore be interpreted wrongly, and is constantly interpreted wrongly (missdeutet). It is applied in differential equations93 as a means of preparing the way for setting x1 - x or h = 0 and of bringing the latter closer to its presentation: - a childishness which has its origin in the first mystical and mystifying methods of calculus. In the application of differential equations to curves, etc., it really serves to make things more apparent geometrically. ___________
Marx's Mathematical Manuscripts 1881