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.
A) Supplement on the differentiation of uz.47
1) For me the essential thing in the last manuscript on the development of d(uz) was the proof, referring to the equation
A) dy/dx = z⋅du/dx + u⋅dz/dx ,
that the algebraic method applied here reverses itself into the differential method, since it develops within the derivative, and thus on the right-hand side, symbolic differential coefficients without corresponding equivalents, real coefficients; hence these symbols as such become independent starting points and ready-made operational formulae.
The form of equation A) lends itself all the more readily to this purpose since it allows a comparison between the du/dx, dz/dx, produced within the derivative f’(x), and the dy/dx, which is the symbolic differential coefficient of f’(x) and therefore comprises its symbolic equivalent, standing opposite on the left-hand side.
Confronting the character of du/dx, dz/dx, as operational formulae, I have been content with the hint that for any symbolic differential coefficient an arbitrary ‘derivative’ may be found as its real value if one substitutes some f(x), 3x² for example, for u and some φ(x), x³ + ax² for example, for z.
I however could also have indicated the geometric applicability of each operational formula, since for example, the general formula for the subtangent of a curve = y⋅dx/dy, which has a form generally identical to z⋅du/dx, u⋅dz/dx, for they are all products of a variable and a symbolic differential coefficient.
Finally, it could have been noted that y = uz [is] the simplest elementary function (y here = y1, and uz is the simplest form of the second power) with which our theme could have been developed.
A) Differentiation of u/z.48
3) Since d(u/z) is the inverse of d(uz), where one has multiplication the other division, one may use the algebraically obtained operational formula
d(uz) = z⋅du + u⋅dz
in order to find d(u/z) directly. I will now do this, in order that the difference between the method of derivation and the simple application of a differential result found previously which now in turn serves as an operational formula, may stand out clearly.
a) y = u/z ,
b) u = yz .
Since y = u/z, thus
yz = (u/z)⋅z = u .
We have thus simply formally concealed u in a product of two factors. Nonetheless, the task is thereby in fact already solved, since the problem has been transformed from the differentiation of a fraction to the differentiation of a product, for which we have the magic formula in our pocket. According to this formula:
c) du = z⋅dy + y⋅dz .
We see immediately that the first term of the second side, namely z⋅dy, must remain sitting in peace at its post until the crack of doom (genau vor Torschluss), since the task consists precisely in finding the differential of y( = u/z ), and thus its expression in differentials of u and z. For this reason, on the other hand, y⋅dz is to be removed to the left-hand side. Therefore:
d) du = y⋅dz = z⋅dy .
We now substitute the value of y, namely u/z, into y⋅dz, so that
du - (u/z)⋅dz = z⋅dy ;
(z⋅du - u⋅dz)/z = z⋅dy .
The moment has now come to free dy of its ‘sleeping partner’* z, and we obtain
(z⋅du - u⋅dz)/z² = dy = d(u/z) .
Marx intended to write several supplements to ‘On the Differential’, four sketches of which survive (for more details see pp.479-490 [Yanovskaya, 1968], which presents a series of extracts from these sketches). Since the drafts are not finished, only two more complete (and understandable) extracts from them are reproduced here. They are adapted from supplements to the second and third drafts.
47 This is Marx’s heading to section A) of the second draft of the supplement to the manuscript ‘On the Differential’. Only point 1), containing a short résumé of the basic work on the differential, is published here. The important supplementary material to the latter work here is the direct indication of the geometric applicability of operational formulae. For more detail see p.479 [Yanovskaya, 1968].
48 This is paragraph A) of the third draft of the supplement. The heading is due to Marx. Published here is only point 3), in which Marx (in his characteristically literary style) introduces the application of the theorem of the differential of a product as an operational formula for finding the derivative of a fraction.
* Original in English - Trans.