Library:On the concept of the derived function

On the concept of the derived function
On the concept of the derived function
Author	Karl Marx
Written in	August 1881
Source	MIA
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I

Let the independent variable $x$ increase to $x_{1}$ ; then the dependent variable $y$ increases to $y_{1}$ .^[1]

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

1) $y = a x$ , when $x$ increases to $x_{1}$ ,

$y_{1} = a x_{1}$ and $y_{1} - y = a (x_{1} - x)$

Now allow the differential operation to occur, that is, we let $x_{1}$ take on the value of $x$ . Then

$x_{1} = x; x_{1} - x = 0$

thus

$a (x_{1} - x) = a \times 0 = 0$

Furthermore, since $y$ , only becomes $y_{1}$ because $x$ increases to $x_{1}$ , we have at the same time

$y_{1} = y; y_{1} - y = 0$

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 $x_{1} - x$ , we then obtain

$\frac{y_{1} - y}{x_{1} - x} = a$

Since $y$ is the dependent variable, it cannot carry out any independent motion at all, $y_{1}$ therefore cannot equal $y$ and $y_{1} - y = 0$ without $x_{1}$ first having become equal to $x$ .

On the other hand we have seen that $x_{1}$ cannot become equal to $x$ in the function $a (x_{1} - x)$ without making the latter $= 0$ . The factor $x_{1} - x$ was thus necessarily a finite difference^[2] when both sides of the equation were divided by it. At the moment of the construction of the ratio

$\frac{y_{1} - y}{x_{1} - x}$

$x_{1} - x$ is therefore always a finite difference. It follows that

$\frac{y_{1} - y}{x_{1} - x}$

is a ratio of finite differences, and correspondingly

$\frac{y_{1} - y}{x_{1} - x} = \frac{Δ y}{Δ x}$

Therefore

$\frac{y_{1} - y}{x_{1} - x}$ or^[3] $\frac{Δ y}{Δ x} = a$ ,

where the constant a represents the limite value (Grenzwert) of the ratio of the finite differences of the variables.^[4]

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

$\frac{y_{1} - y}{x_{1} - x}$ or $\frac{Δ y}{Δ x}$ ,

and this is characteristic of such simple functions as $a x$ .

If in the denominator of this ratio $x_{1}$ decreases so that it approaches $x$ , the limit of its decrease is reached as soon as it becomes $x$ . Here the difference becomes $x_{1} - x_{1} = x - x = 0$ and therefore also $y_{1} - y_{1} = y - y = 0$ . In this manner we obtain

$\frac{0}{0} = a$

Since in the expression $\frac{0}{0}$ every trace of its origin and its meaning has disappeared, we replace it with $\frac{d y}{d x}$ , where the finite differences $x_{1} - x$ or $Δ x$ and $y_{1} - y$ or $Δ y$ appear symbolised as cancelled or vanished differences, or $\frac{Δ y}{Δ x}$ changes to $\frac{d y}{d x}$ .

Thus

$\frac{d y}{d x} = a$ .

The closely-held belief of some rationalising mathematicians that $d y$ and $d x$ are quantitatively actually only infinitely small, only approaching $\frac{0}{0}$ , 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

$y = x$

$y_{1} = x_{1}; y_{1} - y = x_{1} - x$

$\frac{y_{1} - y}{x_{1} - x}$ or $\frac{Δ y}{Δ x} = 1$ ; $\frac{0}{0}$ or $\frac{d y}{d x} = 1$ .

Notes

↑ 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 $x_{1}$ , $y_{1}$ , ...
↑ In keeping with the accepted terminology of the source-books which Marx consulted, a finite difference is here understood always to be a non-zero difference.
↑ Marx distinguishes in each equation two sides (where now we speak of two parts), the left hand and the right hand which do not always play symmetric roles. On the left-hand side of the equation he frequently places two different, equivalent expressions joined by the conjunction ‘or’.
↑ In the mathematical literature which was at Marx’s command the term ‘limit’ (of a function) had no well-defined meaning and was understood most often as the value the function actually reached at the end of an infinite process in which the argument approached its limiting value. Marx devoted an entire rough draft to the criticism of these shortcomings in the manuscript, ‘On the ambiguity of the terms “limit” and “limit value” ‘. In the manuscript before us Marx employs the term ‘limit’ in a special sense: the expression, given by predefinition, for those values of the independent variable at which it becomes undefined. For Marx, the ratios $\frac{Δ x}{Δ y}$ (at $Δ x = 0$ this is transformed to $\frac{0}{0}$ ) and $\frac{d y}{d x}$ , interpreted as the symbolic expression of the ratio ‘of annulled or vanished differences’, that is, of $\frac{0}{0}$ , are such expressions. With respect to the ratio $\frac{Δ y}{Δ x}$ , Marx (influenced to a certain degree by the definitions of this concept in Hind and Lacroix, took this to be an expression which is identically equal to this ratio when $Δ x \neq 0$ , but which has been predefined by continuity when the ratio is transformed to $\frac{0}{0}$ . The ‘limit’, at that point, consequently, must be the ‘preliminary derivative’. Exemplifying this, Marx writes with respect to the ratio $\frac{Δ y}{Δ x}$ where $y = a x^{3} + b x^{2} + c x + d$ : ‘The “preliminary derivative” $a (x_{1}^{2} + x_{1} x + x^{2}) + b (x_{1} + x) + c$ appears here as the limit of a ratio of finite differences; that is, no matter how small we allow the differences to become, the value of $\frac{Δ y}{Δ x}$ will always be given by this “derivative”.’ Later, Marx mentions that setting $x_{1}$ equal to $x$ , that is, setting $Δ x = 0$ , ‘reduces this limit value to its absolute minimum quantity ,’ giving its ‘final derivative’. Analogously, by ‘the limit of the ratio of differentials’ Marx in this manuscript means the ‘real’ (‘algebraic’) expression which provides the value for this ratio; in other words, the derived function. Marx writes, however, that in the equation $\frac{d x}{d y} = f^{'} (x)$ , ‘neither of the two sides is the limiting value of the other. They approach one another, not in a limit relationship, but rather in a relationship of equivalence,’. But here, the concept of ‘limit’ (and of ‘limit value’) is used in another sense, close to the one accepted today. Marx uses the therm ‘absolute minimal expression’ in a sense even closer to the contemporary concept of limit, when he writes in another passage that it is interchangeable with the category of limit, in the sense given it by Lacroix and in which it has had great significance for mathematical analysis.

[1] 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 $x_{1}$ , $y_{1}$ , ...

[2] In keeping with the accepted terminology of the source-books which Marx consulted, a finite difference is here understood always to be a non-zero difference.

[3] Marx distinguishes in each equation two sides (where now we speak of two parts), the left hand and the right hand which do not always play symmetric roles. On the left-hand side of the equation he frequently places two different, equivalent expressions joined by the conjunction ‘or’.

[4] In the mathematical literature which was at Marx’s command the term ‘limit’ (of a function) had no well-defined meaning and was understood most often as the value the function actually reached at the end of an infinite process in which the argument approached its limiting value. Marx devoted an entire rough draft to the criticism of these shortcomings in the manuscript, ‘On the ambiguity of the terms “limit” and “limit value” ‘. In the manuscript before us Marx employs the term ‘limit’ in a special sense: the expression, given by predefinition, for those values of the independent variable at which it becomes undefined. For Marx, the ratios $\frac{Δ x}{Δ y}$ (at $Δ x = 0$ this is transformed to $\frac{0}{0}$ ) and $\frac{d y}{d x}$ , interpreted as the symbolic expression of the ratio ‘of annulled or vanished differences’, that is, of $\frac{0}{0}$ , are such expressions. With respect to the ratio $\frac{Δ y}{Δ x}$ , Marx (influenced to a certain degree by the definitions of this concept in Hind and Lacroix, took this to be an expression which is identically equal to this ratio when $Δ x \neq 0$ , but which has been predefined by continuity when the ratio is transformed to $\frac{0}{0}$ . The ‘limit’, at that point, consequently, must be the ‘preliminary derivative’. Exemplifying this, Marx writes with respect to the ratio $\frac{Δ y}{Δ x}$ where $y = a x^{3} + b x^{2} + c x + d$ : ‘The “preliminary derivative” $a (x_{1}^{2} + x_{1} x + x^{2}) + b (x_{1} + x) + c$ appears here as the limit of a ratio of finite differences; that is, no matter how small we allow the differences to become, the value of $\frac{Δ y}{Δ x}$ will always be given by this “derivative”.’ Later, Marx mentions that setting $x_{1}$ equal to $x$ , that is, setting $Δ x = 0$ , ‘reduces this limit value to its absolute minimum quantity ,’ giving its ‘final derivative’. Analogously, by ‘the limit of the ratio of differentials’ Marx in this manuscript means the ‘real’ (‘algebraic’) expression which provides the value for this ratio; in other words, the derived function. Marx writes, however, that in the equation $\frac{d x}{d y} = f^{'} (x)$ , ‘neither of the two sides is the limiting value of the other. They approach one another, not in a limit relationship, but rather in a relationship of equivalence,’. But here, the concept of ‘limit’ (and of ‘limit value’) is used in another sense, close to the one accepted today. Marx uses the therm ‘absolute minimal expression’ in a sense even closer to the contemporary concept of limit, when he writes in another passage that it is interchangeable with the category of limit, in the sense given it by Lacroix and in which it has had great significance for mathematical analysis.

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