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.

Let the independent variable *x* increase to *x _{1}*; then the dependent variable

here in I) we consider the simplest possible case, where *x* appears only to the first power.

1) *y = ax*, when *x* increases to *x _{1}*,

*y _{1} = ax_{1}* and

Now allow the *differential operation* to occur, that is, we let *x _{1}* take on the value of

*x _{1} = x; x_{1} - x = 0 ,*

thus

*a(x _{1} - x) = a⋅0 = 0 .*

Furthermore, since *y*, only becomes *y _{1}* because

*y _{1} = y ; y_{1} - y = 0 .*

Thus

*y _{1} - y = a(x_{1} - x)*

changes to 0 = 0.

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(x _{1} - x)* and the left side of the corresponding equation by the factor

*(y _{1} - y)/(x_{1} - x) = a .*

Since *y* is the *dependent variable*, it cannot carry out any independent motion at all, *y _{1}* therefore cannot equal

On the other hand we have seen that *x _{1}* cannot become equal to

*(y _{1} - y)/(x_{1} - x)*

*x _{1} - x* is therefore always a finite difference. It follows that

*(y _{1} - y)/(x_{1} - x)*

is a *ratio of finite differences*, and correspondingly

*(y _{1} - y)/(x_{1} - x) = Δy/Δx*

Therefore

*(y _{1} - y)/(x_{1} - x)* or

where the constant a represents the *limite value (Grenzwert)* of the ratio of the finite differences of the variables.^{5}

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

*(y _{1} - y)/(x_{1} - x)* or

and this is characteristic of such simple functions as *ax*.

If in the denominator of this ratio *x _{1}* decreases so that it approaches

*0/0 = a .*

Since in the expression *0/0* every trace of its origin and its meaning has disappeared, we replace it with *dy/dx* , where the finite differences *x _{1} - x* or

Thus

*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 *

*y _{1} = x_{1} ; y_{1} - y = x_{1} - x ;*

*(y _{1} - y)/(x_{1} - x)* or

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 = ax³ + bx² + cx - e* .

If *x* increases to *x _{1}*, then

*y _{1} = ax_{1}³ + bx² + cx_{1} - e ,*

*y _{1} - y = a(x_{1}³ - x³) + b(x_{1}² - x²) + c(x_{1} - x)*

*= a(x _{1} - x) (x_{1}² + x_{1}x + x²)*

*+ b(x _{1} - x) (x_{1} + x) + c(x_{1} - x) .*

Therefore

*(y _{1} - y)/(x_{1} - x)* or

and the the *preliminary ‘derivative’* [is]

*a(x _{1}² + x_{1}x + x²) + b(x_{1} + 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 *Δy/Δx* 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 x_{1} is decreased in the function

*a(x _{1}² + x_{1}x + x²) + b(x_{1} + x) + c*

until it has reached the limit of its decrease, that is, has become *the same as x*, [then] *x _{1}²* is changed to

*3ax² + 2bx + c .*

It Is here shown in a striking manner:

*First*: in order to obtain the ‘derivative’, *x _{1}* must be set

*Second*: Although we set *x _{1} = x* and therefore

The reduction of *x _{1}* to

*0/0* or *dy/dx = 3ax² + 2bx + c ,*

so that the *derivative* appears as the *limit value* of the ratio of the differentials.

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’

*3ax² + 2bx + c*

the variable *x* exists in a completely different condition than in the original function of *x* (namely, in *ax³ + bx² + cx - e*). It [this derivative] can therefore itself be treated as an original function in turn, and can become the mother of another ‘derivative’ by the repeated process of differentiation. This can be repeated as long as the variable *x* has not been finally removed from one of the ‘derivatives’; it therefore continues endlessly in functions of *x* can only be represented by infinite series, which [is] all to often the case.

The symbols *d²y/dx² , d³y/dx³,* 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 *3x²*, for example, can pass through the process of differentiation as well as its mother *x³*. One could just as well begin with *3x²* as with the original function of *x*.

But *nota bene*. The starting point of the process of differentiation actually is Δy/Δx 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]:

*Δy/Δx = a = dy/dx .*

Here therefore as a matter of fact *no new limit value* is found from the process of differentiation which *Δy/Δx* 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, d²y/dx²,* 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 *x _{1}*, then

*y _{1} - y = a(x_{1}^{m} - x^{m})*

*= a(x _{1} - x) (x_{1}^{m-1} + x_{1}^{m-2}x + (x_{1}^{m-3}x² +* etc.

*up to the term x _{1}^{m-m}x^{m-1}).*

Therefore

*(y _{1} - y)/(x_{1} - x)* or

*+ x _{1}^{m-m}x^{m-1}).*

We now apply the process of differentiation to this *‘preliminary derivative’*, so that

*x _{1} = x* or

and

*x _{1}^{m-1}* is change into

*x _{1}^{m-2}x* into

*x _{1}^{m-3}x²* into

and finally,

*x _{1}^{m-m}x^{m-1}* into

We thus obtain the function *x ^{m-1}m* times, and the ‘derivative’ is therefore

Due to the equivalence of *x _{1} = x* within the ‘preliminary derivative’,

*dy/dx = max ^{m-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 *x _{1} - x* appeared

*(y _{1} - y)/(x_{1} - x)* or

This [is] not the case in the following:

3) *y = a ^{x}* ;

Let *x* become *x _{1}*. Then

*y _{1} = a^{x1}*

Therefore

*y _{1} - y = a^{x1} - a^{x} = a^{x}(a^{x1-x} - 1) .*

[But]

*a ^{x1 - x} = {1 + (a - 1)}^{x1-x} ,*

and

*{1 + (a - 1)} ^{x1 - x} =*

*1 + (x _{1} - x) (a - 1) + ((x_{1} - x) (x_{1} - x - 1) (a - 1)²)/1⋅2 + etc.^{8}*

Therefore

*y _{1} - y = a^{x} (a^{x1 - x} - 1)*

*= a ^{x}⋅{(x_{1} - x)(a - 1) + ((x_{1} - x) (x_{1} - x - 1) (a - 1)²)/1⋅2 + ((x_{1} - x)(x_{1} - x - 1) (x_{1} - x - 2) (a - 1)³)/1⋅2⋅3 + etc.}*

∴^{9}

*(y _{1} - y)/(x_{1} - x)* or

*a ^{x}⋅{(a - 1) + ((x_{1} - x - 1) (a - 1)²)/1⋅2 + ((x_{1} - x - 1) (x_{1} - x - 2) (a - 1)³)/1⋅2⋅3 + etc.}.*

Now as *x _{1} = x* and thus

*a ^{x}⋅{(a - 1) - (a - 1)²/2 + (a - 1)³/3 - etc.}.*

Thus

*dy/dx = a x⋅{(a - 1) - (a - 1)²/2 + (a - 1)³/3 - etc.}*

If we designate the sum of the constants in parentheses A, then

*dy/dx = Aa ^{x} ;*

but this A = Napierian logarithm of the number^{*5} *a*, so that:

*dy/dx*, or, when we replace *y* by its value: *da ^{x}x>/dx = log a⋅a^{x} ,*

and

*da ^{x} = log a⋅a^{x}dx .*

Supplementary^{10}

We have considered

1) cases in which the factor *(x _{1} - x)* [occurs] only once in [the expression which leads to] the

*(y _{1} - y)/(x_{1} - x)* or

this same factor is therefore eliminated from the function of *x*.

2) ( in the example *d(a ^{x})*) cases in which factors of

3) Yet to be considered is the case where the factor *x _{1} - x* is

*y = sqrt{a² + x²},*

*y _{1} = sqrt{a² + x_{1}²},*

*y _{1} - y = sqrt{a² +x_{1}²} - sqrt{a² + x²};*

we divide the function of *x*, the left-hand side as well, therefore, by *x _{1} - x*. Then

*(y _{1} - y)/(x_{1} - x) (*or

In order to rationalise the numerator, [both] numerator and denominator are multiplied by *sqrt{a² + x _{1}²} + sqrt{a² + x²}*, and we obtain:

*Δy/Δx = (a² + x _{1}² - (a² + x²))/((x_{1} - x)(sqrt{a² + x_{1}²} + sqrt{a² + x²})*

*= (x _{1}² - x²)/((x_{1} - x)(sqrt{a² + x_{1}²} + sqrt{a² + x²}))*

But

(x_{1}² - x²)/((x_{1} - x)(sqrt{a² + x_{1}²} + sqrt{a² + x²}))

= ((x_{1} - 1)(x_{1} + x))/((x_{1} - x)(sqrt{a² + x_{1}²} + sqrt{a² + x²})) .

So that:

*Δy/Δx = (x _{1} + x)/(sqrt{a² + x_{1}²} + sqrt{a² + x²})*

Now when *x _{1}* becomes

*dy/dx = 2x/(2⋅sqrt{a² + x²}) = x/sqrt{a² + x²}*

So that

*dy* or *d sqrt{a² + x²} = xdx/sqrt{a² + x²} .*

^{1}
The manuscript was written in 1881 for Engels. This is the first work in a series of manuscripts conceived by Marx and devoted to a systematic exposition of his ideas on the nature and history of differential calculus. In this work he introduces his concepts of algebraic differentiation and the corresponding algorithm for finding the derivative for certain classes of functions. On the envelope enclosing the manuscript there is the notation in Marx’s handwriting: ‘For the General’. This was Engels’s nickname in Marx’s family because of his articles on military questions. Having acquainted himself with the manuscript, Engels answered Marx in a letter on 18 August 1881 (see p.xxvii). The published German text of the manuscript reproduces exhaustively Marx’s text. Some of the preparatory material (drafts and supplements) is published on page 473 of Yanovskaya, 1968. Variant readings from the unpublished drafts are provided in footnotes. The manuscript was published for the first time (not in full) in 1933 in Russian translation in the collection Marxism and Science (Marksizm I estestvoznanie), Moscow, Partizdat, 1933, pp.5-11; and in the journal Under the Banner of Marxism (Pd znamenem marksizma) No.1, 1933, p.15ff. This is the first time it has been published in German.

^{2}
In order to avoid confusion with the designation of derivatives, Marx’s notation *x´, y´*, ... for the new values of the variable has been replaced here and in all similar cases by *x _{1}, y_{1},* ...

Original: ‘root’. -