Intelligent Design

Updating the Second Derivative

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In calculus, the Leibniz notation for the second derivative confuses most students. It turns out, rightfully so. The notation for the second derivative turns out to actually be incorrect.

Earlier this year, I got a paper published detailing the problem and the corrected notation. In the video below, I introduce the new notation, why the old notation is problematic, how the new notation can be derived straightforwardly, and why it may matter in the future.

For those interested, the problems with the notation for the second derivative are well-known, and a kludge exists for working around it known as Faa di Bruno’s formula (a simplified version being known as “the chain rule for the second derivative”). However, this formula does not work out algebraically with the differentials. In the new notation, all of the differentials work together algebraically in a way that was not generally thought to be possible before.

For those interested, I have other (less radical) thoughts for improving the way that calculus is thought about and taught. You can also see them in action in my book Calculus from the Ground Up (shameless plug alert).

7 Replies to “Updating the Second Derivative

  1. 1
    DerekDiMarco says:

    I suspect students opening the 37th edition of Stewart in 2056 will still see d²y/dx².

  2. 2
    redwave says:

    johnnyb. You express a bit of courageousness in publishing your personal confusion in the treatment of the calculus. Yet your video does not offer clarity for understanding the second derived function. Possibly the confusion is rooted in a requirement for “notation” to be treated algebraically when that which is noted can be treated algebraically with little to no confusion … otherwise noting the derived function of a function is a derived function such that with a physical example that acceleration is the derived function of speed which is the derived function of distance, to wit the second derivative however noted as d2y/dx2 or y” or f”(x).

    I would suggest you and your students begin with Calculus, an Intuitive and Physical Approach, by Morris Kline (1967,1977) to understand the concepts of a function, the derivative (the derived function), and the method of increments.

  3. 3
    johnnyb says:

    Redwave –

    I didn’t follow what you were saying exactly. The fact is that the second derivative *can’t* be utilized algebraically, and the video gives an example of what happens when you do. This part is well-known and is often mentioned at least in passing in most calculus texts.

    My point is that the *reason* it can’t be utilized algebraically is because it is *incorrect*. I say it is incorrect because (a) it is written as a fraction, (b) it can’t be used as a fraction, and (c) there exists a way of writing it as a fraction such that it *can* be used as a fraction. Perhaps we could say that “it’s only notation” or something similar if there *were* no correct way of writing it as a usable fraction. Since there is, I think it would be valid to say that the normal way of doing it is incorrect.

    I don’t think I’ve read Kline’s calculus, though I have read many others. My calculus textbook comes more from the perspective on non-standard analysis, though its focus is on communicating (a) the intuition of calculus, and (b) teaching students how to use calculus to build new formulas.

    Calculus from the Ground Up

  4. 4
    johnnyb says:

    DerekDiMarco –

    You are probably correct. The most common response I’ve received from mathematicians about this is “you’re not wrong, but I don’t like it”.

  5. 5
    redwave says:

    johnnyb. I apologise that my brief response was not clear. I also apologize for suggesting that I would teach you mathematics. We most likely have differing approaches and also approach mathematics with differing philosophies. I attempted some clarity as follows: The notation for the second derived function is rather confusing in style and so I would suggest again that Kline’s approach might help. The second derived function is similar to the first in terms of a ratio, dy/dx , not a fraction and instead of listing 1st dy/dx, 2nd dy/dx, 3rd dy/dx in where one repeats upon the results or y=x^3, 1st dy=3x^2, 2nd dy=6x, 3rd dy=6 et cetera … and the dx changes at each step because y and x are variables in relationship here. The conventional notation is calculus not algebra and is not meant to be treated as algebraic terms. dy/dx is more often considered d/dx(x^n) as one first derived function. dy/dx is not a fraction, rather the symbol for an operation to be performed (J.C. Burkill, A first course in Mathematical Analysis). If y=x^3 is the function to be considered then the ratio of y to x would be dy/dx(x^n) = nx^n-1 or 3x^2 and the derivation is repeated until one derives a constant or zero. This is only one example of a derivative. Others exist for multiple functions, such as trigonometric, et cetera. dy/dx is not a fraction for infinites/infinitesimals alone or exclusively, or one might say infinitesimals are not necessarily infinitely small in an absolute sense of infiniteness. And limits are not restricted in a similar way. I have suggested that you show a bit of courageousness especially in attempting to algebraicize the calculus, yet this process is not necessary in establishing a thorough analysis. I would recommend, Unknown Quantity – A real and imaginary history of algebra by John Derbyshire for understanding algebra in the broader context of mathematics.

  6. 6
    johnnyb says:

    Redwave –

    Here’s the question – why the move away from Algebra? Calculus was originally based on the algebra of infinitesimals. Why the move? The answer is that in the 1800s it was thought that infinitesimals were non-rigorous, largely from the bias towards physicalism of the 19th century mathematicians (infinities and infinitesimals are often too close to idealism for the comfort of materialists).

    What advantage does removing the algebraic properties yield? If it can be defined in such a way as to maintain the algebraic properties, what is the benefit of not doing so?

    With the fraction/ratio notation, the question is even deeper – there are innumerable ways to represent a function of two variables. Why choose dy/dx if you don’t actually mean a ratio? Arbogast had a decent notation of D_x(y). The reason for dy/dx was *because* it was originally conceived as a ratio.

    So, rather than deal with the difficulties, they just defined them away. But how is that helpful? There is an algebraically-manipulable notation available, which is obtainable simply by applying the quotient rule to….a quotient. What advantage is there to not doing it?

    The advantage of the notation I have proposed is the ability to ask additional questions. I can’t tell you where they will lead, if anywhere at all. For instance, in this notation, it is a perfectly legitimate question to ask, what is “d^2y/d^2x” (notice where the “2” is on the bottom). This is nonsense in the “operator” sense, but a perfectly sensible question in the algebraic sense.

    If nothing else, the algebraic approach (a) leverages existing knowledge that students learning calculus have, (b) simplifies the mechanisms of calculus, and (c) unifies a lot of operations of calculus. See Simplifying and Refactoring Introductory Calculus for information on this. When treated algebraically, Calculus is actually very straightforward to most students. And I don’t see even one thing that is lost.

  7. 7
    redwave says:

    johnnyb. Thank you for responding to my comments/observations. I might have merely added to the confusion and your method is not intrinsically mistaken. Possibly the difference between us is wrapped up in the notation and not in what follows. “More power to you” in simplifying the Calculus, so that others understand. I see no benefit in attempts to convince you concerning my perspective, as yours has helped you with the Calculus.

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