Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. And yet it is clear that dy/dx can only be the pure expression of a completed process if the last trace of the quanta x and y has disappeared, leaving the expression of the preceding process of their change without any quantity. You need not fear that any mathematician has preceded you here. This kind of differentiation is indeed much simpler than all others, so that just now I applied it myself to derive a formula I had suddenly lost, confirming it afterwards in the usual way. The procedure must have made the greatest sensation, especially, as is clearly proved, since the usual method of neglecting dxdy etc. is positively false. And that is the special beauty of it: only if dy/dx = 0/0 is the mathematical operation absolutely correct. So old Hegel guessed quite correctly when he said that differentiation had for its basic condition that the variables must be raised to different powers, and at least one of them to at least the second, or power. Now we also know why. If we say that in y = f(x) the x and y are variables, then this claim has no further consequences, as long as we do not move on, and x and y are still, pro tempore, in fact constants. Only when they really change, i.e. inside the function, do they indeed become variables, and only then can the relation still hidden in the original equation reveal itself not the relation of the two magnitudes but of their variability. The first derivative Dy/Dx shows this relation as it happens in the course of real change, i.e. in each given change; the completed derivative dy/dx shows it in its generality, pure, and hence we can come from dy/dx to each Dy/Dx, while the latter itself only covers the special case. However, to pass from the special case to the general relationship, the special case must be abolished (aufgehoben) as such. Hence, after the function has passed through the process from x to x with all its consequences, x can be allowed calmly to become x again; it is no longer the old x, which was variable in name only; it has passed through actual change, and the result of the change remains, even if we again abolish (aufheben) it. At last we see clearly what mathematicians have claimed for a long time, without being able to present rational grounds, that the differential-quotient is the original, the differentials dx and dy are derived: the derivation of the formulae demands that both so-called irrational factors stand at the same time on one side of the equation, and only if you put the equation back into this its first form dy/dx = f'(x) , as you can see, are you free of the irrationals and instead have their rational expression. The thing has taken such a hold of me that it not only goes round my head all day, but last week in a dream I gave a chap my shirt buttons to differentiate, and he ran off with them.
Letters: Marx-Engels Correspondence 1881