Let the independent variable x increase to x1; then the dependent variable y increases to y1.2 here in I) we consider the simplest possible case, where x appears only to the first power. y1 = ax1 and y1 - y = a(x1 - x) Now allow the differential operation to occur, that is, we let x1 take on the value of x. Then x1 = x; x1 - x = 0 , a(x1 - x) = a 0 = 0 . Furthermore, since y, only becomes y1 because x increases to x1, we have at the same time y1 = y ; y1 - y = 0 . y1 - y = a(x1 - x) First making the differentiation and then removing it therefore leads literally to nothing. The whole difficulty in understanding the differential operation (as in the negation of the negation generally) lies precisely in seeing how it differs from such a simple procedure and therefore leads to real results. If we divide both a(x1 - x) and the left side of the corresponding equation by the factor x1 - x, we then obtain (y1 - y)/(x1 - x) = a . Since y is the dependent variable, it cannot carry out any independent motion at all, y1 therefore cannot equal y and y1 - y = 0 without x1 first having become equal to x. On the other hand we have seen that x1 cannot become equal to x in the function a(x1 - x) without making the latter = 0. The factor x1 - x was thus necessarily a finite difference3 when both sides of the equation were divided by it. At the moment of the construction of the ratio (y1 - y)/(x1 - x) (y1 - y)/(x1 - x) (y1 - y)/(x1 - x) = y/ x (y1 - y)/(x1 - x) or4 y/ x = a , Since a is a constant, no change may take place in it; hence none can occur on the right-hand side of the equation, which has been reduced to a. Under such circumstances the differential process takes place on the left-hand side (y1 - y)/(x1 - x) or y/ x , If in the denominator of this ratio x1 decreases so that it approaches x, the limit of its decrease is reached as soon as it becomes x. Here the difference becomes x1 - x1 = x - x = 0 and therefore also y1 - y = y - y = 0. In this manner we obtain 0/0 = a . dy/dx = a . The closely-held belief of some rationalising mathematicians that dy and dx are quantitatively actually only infinitely small, only approaching 0/0, is a chimera, which will be shown even more palpably under II). As for the characteristic mentioned above of the case in question, the limit value (Grenzwert) of the finite differences is therefore also at the same time the limit value of the differentials. 2) A second example of the same case is y = x y1 = x1 ; y1 - y = x1 - x ; (y1 - y)/(x1 - x) or y/ x = 1 ; 0/0 or dy/dx = 1 . When in y = f(x) , the function [of] x appears on the right-hand side of the equation in its developed algebraic expression,6 we call this expression the original function of x, its first modification obtained by means of differentiation the preliminary derived function of x and its final form obtained by means of the process of differentiation the derived function of x.7 1) y = ax + bx + cx - e . If x increases to x1, then y1 = ax1 + bx + cx1 - e , y1 - y = a(x1 - x ) + b(x1 - x ) + c(x1 - x) = a(x1 - x) (x1 + x1x + x ) + b(x1 - x) (x1 + x) + c(x1 - x) . Therefore (y1 - y)/(x1 - x) or y/ x = a(x1 + x1x + x ) + b(x1 + x) + c . a(x1 + x1x + x ) + b(x1 + x) + c [and it] is here the the limit value (Grenzwert) of the ratios of the finite differences; that is, however small these differences may become, the value of y/ x is given by that derivative . But this is not the same case as that under I) with the limit value of the ratios of the differentials.* When the variable x1 is decreased in the function a(x1 + x1x + x ) + b(x1 + x) + c 3ax + 2bx + c . It Is here shown in a striking manner: First: in order to obtain the derivative , x1 must be set = x; therefore in the strict mathematical sense x1 - x = 0, with no subterfuge about merely approaching infinitely [closely]. Second: Although we set x1 = x and therefore x1 - x = 0, nonetheless nothing symbolic appears in the derivative .*2 The quantity x1, although originally obtained from the variation of x, does not disappear; it is only reduced to its minimum limit value = x. It remains in the original functions of x as a newly introduced element which, by means of its combinations partly with itself and partly with the x of the original function, finally produces the derivative , that is, the preliminary derivative reduced to its absolute minimum quantity. The reduction of x1 to x within the first (preliminary) derived function changes the left-hand side [from] y/ x to 0/0 or dy/dx, thus: 0/0 or dy/dx = 3ax + 2bx + c , The transcendental or symbolic mistake which appears only on the left-hand side has perhaps already lost its terror since it now appears only as the expression of a process which has established its real content on the right-hand side of the equation. In the derivative 3ax + 2bx + c The symbols d y/dx , d y/dx , etc., only display the genealogical register of the derivatives with respect to the original given function of x. They are mysterious only so long as one treats them as the starting point of the exercise, instead of a merely the expression of the successively derived functions of x. For it indeed appears miraculous that a ratio of vanished quantities should pass through a new hight degree of disappearance, while there is nothing wonderful in the fact that 3x , for example, can pass through the process of differentiation as well as its mother x . One could just as well begin with 3x as with the original function of x. But nota bene. The starting point of the process of differentiation actually is y/ x only in equations as [above] under I), where x appears only to the first power. Then, however, as was shown under I), the result [is]: y/ x = a = dy/dx . Here therefore as a matter of fact no new limit value is found from the process of differentiation which y/ x passes through; [a result] which remains possible only so long as the preliminary derivative includes the variable x, so long, therefore, as dy/dx remains the symbol of a real process.*3 Of course, it is in no sense an obstacle, that in the differential calculus the symbols dy/dx, d y/dx , etc., and their combinations also appear on the right-hand side of the equation. For one knowns as well that such purely symbolic equations only indicate the operations which are then to be applied to the real function of variables. 2) y = axm . As x becomes x1, then y1 = ax1m and = a(x1 - x) (x1m-1 + x1m-2x + (x1m-3x + etc. up to the term x1m-mxm-1). Therefore + x1m-mxm-1). We now apply the process of differentiation to this preliminary derivative , so that x1 = x or x1 - x = 0 x1m-1 is change into xm-1; x1m-2x into xm-2x = xm-2+1 = xm-1 ; x1m-3x into xm-3x = xm-3+2 = xm-1 , x1m-mxm-1 into xm-mxm-1 = x0+m-1 = xm-1 . We thus obtain the function xm-1m times, and the derivative is therefore maxm-1. Due to the equivalence of x1 = x within the preliminary derivative ,*4 on the left-hand side y/ x is changed to 0/0 or dy/dx; therefore dy/dx = maxm-1 . All of the operations of the differential calculus could be treated in this manner, which would however be a damned useless mass of details. Nonetheless here is another example; since up to now the difference x1 - x appeared only once in the function of x and therefore disappeared from the right-hand side by means of the formation of (y1 - y)/(x1 - x) or y/ x . This [is] not the case in the following: 3) y = ax ; Let x become x1. Then y1 = ax1 y1 - y = ax1 - ax = ax(ax1-x - 1) . ax1 - x = {1 + (a - 1)}x1-x , and Therefore = ax {(x1 - x)(a - 1) + ((x1 - x) (x1 - x - 1) (a - 1) )/1 2 + ((x1 - x)(x1 - x - 1) (x1 - x - 2) (a - 1) )/1 2 3 + etc.} ax {(a - 1) + ((x1 - x - 1) (a - 1) )/1 2 + ((x1 - x - 1) (x1 - x - 2) (a - 1) )/1 2 3 + etc.}. Now as x1 = x and thus x1 - x = 0, we obtain for the derivative : ax {(a - 1) - (a - 1) /2 + (a - 1) /3 - etc.}. Thus dy/dx = ax {(a - 1) - (a - 1) /2 + (a - 1) /3 - etc.} If we designate the sum of the constants in parentheses A, then dy/dx = Aax ; but this A = Napierian logarithm of the number*5 a, so that: dy/dx, or, when we replace y by its value: daxx>/dx = log a ax , dax = log a axdx . Supplementary10 We have considered 1) cases in which the factor (x1 - x) [occurs] only once in [the expression which leads to] the preliminary derivative - i.e. [in] the equation of finite differences11 - so that by means of the division of both sides by x1 - x in the formation of (y1 - y)/(x1 - x) or y/ x 2) ( in the example d(ax)) cases in which factors of (x1 - x) remain after the formation of y/ x.12 3) Yet to be considered is the case where the factor x1 - x is not directly obtained from the first difference equation ([which leads to] the preliminary derivative ). y = sqrt{a + x }, y1 = sqrt{a + x1 }, y1 - y = sqrt{a +x1 } - sqrt{a + x }; (y1 - y)/(x1 - x) (or y/ x) = (sqrt{a + x1 } - sqrt{a + x })/(x1 - x) . In order to rationalise the numerator, [both] numerator and denominator are multiplied by sqrt{a + x1 } + sqrt{a + x }, and we obtain: y/ x = (a + x1 - (a + x ))/((x1 - x)(sqrt{a + x1 } + sqrt{a + x }) = (x1 - x )/((x1 - x)(sqrt{a + x1 } + sqrt{a + x })) y/ x = (x1 + x)/(sqrt{a + x1 } + sqrt{a + x }) Now when x1 becomes = x, or x1 - x = 0, then dy/dx = 2x/(2 sqrt{a + x }) = x/sqrt{a + x } dy or d sqrt{a + x } = xdx/sqrt{a + x } .
Marx's Mathematical Manuscripts 1881