Marx's Mathematical Manuscripts 1881

Written: August, 1881;

Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;

First published: in Russian translation, in *Pod znamenem marksizma*, 1933.

*c) Continuation of [p.] 25 ^{*}*

We have *x _{1} - x = Δx* from the beginning for the expression of the

The simple original setting *x _{1} - x = Δx =* anything

It’s just the same in *x _{1} - x = Δx, x_{1} - Δx = x*. We have the difference form again here on the left-hand side, but this time as the difference between the increased

If however one leaves the mystical differential calculus, where *x _{1} - x* enters immediately as

*d)* From this simple distinction of form there immediately results a fundamental difference in the treatment of the calculus which we demonstrate it detail (see the enclosed loose sheets)^{72} in the analysis of d’Alembert’s method. Here we have only to remark in general:

1) If the *difference x _{1} - x* (and thus

2) Since in the general form *x _{1} = x + Δx* the difference

3) If on the other hand we consider the development of the function in *x* itself, the binomial theorem then gives us for this first term, here *x*, the series of its derived functions. For example, if we have *(x + h) ^{4}*, where

*x ^{4} + 4x³h + etc.*

*4x³*, which appears in the second term and has the factor h raised to the first power, is thus the first derived function of *x*, or, expressed algebraically: if we have *(x + h) ^{4}* as the

*f(x + Δx) = (x + Δx) ^{4} = x^{4} + 4x³Δx +* etc.

*x ^{4}*, which is provided for us in the usual algebraic binomial

4) Furthermore: the second term of the binomial expansion, *4x³h*, provides us immediately *ready-made (fix und fertig)* with the first derived function of x^{4}, namely *4x³*. Thus this derivation has been obtained by the expansion of

*f(x + Δx) = (x + Δx) ^{4} ;*

obtained by means of the interpretation from the beginning of the difference *x _{1} - x* as its

It is thus the binomial expansion of *f(x + Δx)*, or *y _{1}*, which

The crucial point *(Angelpunkt)* of this method is thus the development from the undefined expression *y _{1}* or

5) The only difference equation which comes out in this method is the one which we obtain immediately:

*f(x + Δx) = (x + Δx) ^{4} = x^{4} + 4x³Δx + 6x²Δx² + 4xΔx³ + Δx^{4},*

when we write:

*x ^{4} + 4x³Δx + 6x²Δx² + 4xΔx³ + Δx^{4} ,*

putting the original function *x ^{4}*, which forms the beginning of the series, back again behind, we now have before us the

*4x³Δx + 6x²Δx² + 4xΔx³ + Δx ^{4} ,*

the increment of the original function, *x ^{4}*. This way we use, on the other hand,

*y* or *f(x) = x ^{4} .*

So that Newton also writes immediately:

*dy, to him y. = 4x³x. +* etc.

6) The entire remaining development now consists of the fact that we have to liberate the ready-made derivative *4x³* from its factor *Δx* and from its neighbouring terms, to prise it loose from its surroundings. So this is no method of development, but rather a *method of separation*.

*e) The differentiation of f(x) (as [a] general expression)*

Let us note first of all *(d’abord)* that the concept of the ‘derived function’, for the successive real equivalents of the symbolic differential coefficients, which was completely unknown to the original discoverers of differential calculus and their first disciples, was in fact first introduced by Lagrange. To the former the dependent variable, *y* for example, appears only as a *function of x*, corresponding completely to the original algebraic meaning of *function*, first applied to the so-called indeterminate equations where there are more unknowns than equations, where therefore *y*, for example, takes on different values as different values are assumed for *x*. With Lagrange, however, the original function is the defined expression of *x* which is to be differentiated; so if *y* or *f(x) = x ^{4}*, the

In the algebraic method, where we first develop *f ^{1}*, the preliminary derivative or [the ratio of] finite differences, and where we first develop from it the definitive derivative,

We find

*f ^{1}(x) = Δy/Δx,* so that

And so as well:

*Δy = f ^{1}(x)Δx ,*

and since *Δy = Δf(x)*,

*Δf(x)= f ^{1}(x)Δx .*

The next development of the differential expression, which finally yields

*df(x) = f’(x)dx ,*

is simply the differential expression of the previously developed finite difference.

In the usual method

*dy* or *df(x) = f’(x)dx*

is not developed at all, rather instead, see above, the *f’(x)* provided ready-made by the binomial *(x + Δx)* or *(x + dx)* is only *separated* from its factor and its neighbouring terms.

^{*}
See p.84.

^{†}
In English in the original.

^{72}
This refers to rough-draft notes, divided into sections, part of which are published in this edition under the general heading ‘First Draft’. See pp.76-90 of this edition.

^{*2}
Original *d’abord - Trans*.