A Different Take on Differentiation

A Different Take on Differentiation

I recently came across a tweet with a screenshot from a calculus book that defines the derivative with no mention of a limit. Not even Leibniz’s differentials have a home here. And although the definition is quite verbose, I found that it makes the derivative feel natural — tangible even.

If you’ve taken a calculus course in the last 120 years or so, then you’ve undoubtedly encountered the concept of a limit. In fact, a quick search for the question “Why are limits important?” returns hundreds, if not thousands, of results stating that limits are an essential part of calculus. How could you possibly understand the subject without them?

It turns out that you can develop a great deal of calculus — at least the entire first-year curriculum — without limits. And doing so, in my opinion, provides a deeper understanding of the subject, its history, and a better appreciation for limits and why they are useful.

The Problem With Limits

While statements professing that you can’t do calculus without limits surely have Leibniz turning in his grave, it’s not surprising that students come away with this mindset. Limits are everywhere in the calculus curriculum.

Limits are abstract, though, and students who struggle to understand them will likely struggle with all of calculus. Some students will be doomed to failure, which is a tragedy considering Newton had little more than an intuition about limits. It took nearly a century after Newton for Cauchy and Weierstrass to formalize limits into the ϵ-δ definition used today.

Definition. Let f(x) be a function defined on an open interval around a (note that f(a) need not be defined). Then we say that the limit of f(x) as x approaches a is L and write

$$ \lim_{x \to a} f(x) = L $$

if for every number ϵ > 0 there exists some number δ > 0 such that

$$ |f(a) - L| < \epsilon \hspace{6pt} \textrm{whenever} \hspace{6pt} 0 < |x - a| < \delta $$

What makes this definition difficult to understand? In my opinion, the reasons are twofold:

  1. Quantifier complexity: Most first-year calculus students have no exposure to propositional logic, much less statements with multiple quantifiers whose order is important. Unpacking this logic — some of which is literally written in Greek — is a tall order.
  2. “Circular” reasoning: There’s nothing logically inconsistent with the ϵ-δ definition of a limit, but using the definition can feel circular. To show that the limit of as x → 2 is 4, you must guess that the limit is 4. Then you have to find a suitable δ in terms of ϵ, but proofs that omit how δ is determined feel magical. And not the fun kind of magic, mind you.

Math educators, aware of students’ difficulties with limits, have endured their own struggle to invent better ways of teaching the topic. In 1981, Jerrold Marsden and Alan Weinstein, professors at the University of California, Berkely, proposed a new method — don’t teach limits.

Calculus Unlimited

Calculus without limits isn’t new. Gottfried Leibniz devised calculus using differentials, which are infinitesimal positive quantities less than any real number. The method of exhaustion, developed independently by the ancient Greeks and the Chinese, can be used to find areas and volumes of round shapes, like circles and cones.

Marsden and Weinstein took the latter approach and adapted the method of exhaustion to differentiation. In the preface to their book Calculus Unlimited, on the page screenshotted in the tweet that caught my attention, the authors present instructors with the idea of overtaking [1].

Definition. Let f and g be real-valued functions with domains contained in ℝ, and let a be a real number. We say that f overtakes g at a if there is an open interval I containing a such that

  1. x ∈ I and x ≠ a implies x is in the domain of f and g.
  2. x ∈ I and x < a implies f(x) < g(x).
  3. x ∈ I and x > a implies f(x) > g(x).

There’s a lot of notation in that definition, but the idea lends itself to a natural geometric interpretation.

For example, draw the graphs of two functions f and g that intersect at some point whose x-coordinate is a. If, around some interval I on the x-axis, the graph of f is below the graph of g to the left of a above the graph of g to the right of a, then f overtakes g at a.

To define the derivative, Marsden and Weinstein look at the slopes of lines through a.

Definition: Let f be a function defined on an open interval containing a. Consider the family of lines lₘ(x) = f(a) + m(x-a). Suppose there is a number b such that

  1. m < b implies f overtakes lₘ at a.
  2. m > b implies lₘ overtakes f at a.

Then we say that f is differentiable at a, and that b is the derivative of f at a.

Like the definition of overtaking, Marsden and Weinstein’s definition of the derivative paints a convenient geometric picture. It even provides a tangible method to find the derivative of a function at some point using graphing software.

For instance, consider the graph of f(x) = x² and some point, say a = 1. Graph any line that passes through f(a) = 1² = 1. Now vary the slope of the line until you find a new line that neither overtakes f nor is overtaken by f. If you can find such a line, its slope is the derivative of f(x) at a.

If you play around with this method in graphing software long enough, you’ll notice that you need to zoom in very close to the point f(a) to get any semblance of precision. I found this to be a beautiful and natural manifestation of what the ϵ-δ definition of a limit represents — namely the concern with points on f arbitrarily close to the limit — but manages to obscure from students through its cryptic formulation.

This visual method for finding the derivative at a point isn’t rigorous. It also lacks precision for all but the nicest of functions at the nicest of points. It does emphasize something, however, that was lost on me as a student for a long time: the tangent line is special.

The Derivative as a Transition Point

In chapter 2 of Marsden and Weinstein’s textbook, the authors introduce the notion of a transition point as a point at which something suddenly changes. There are countless examples of transition points in nature: sunrise is the transition point from day to night, 100 degrees celsius is the point at which liquid water transitions to water vapor (at sea level, anyway).

The tangent line can be thought of as the transition point between the set of all lines through a point a that are overtaken by f and the set of all lines through a that overtake f. Something special happens with the tangent line that splits the set of all lines through the point a into two disjoint sets.

Transition points are perhaps the foundational concept in Calculus Unlimited. Marsden and Weinstein frame integrals through the lens of transition points, as well, and the concept plays an important role in their proof of The Fundamental Theorem of Calculus.

The New is Old

Marsden and Weinstein mention in their preface that “as far as [they] know, [their] definition [of the derivative] has not appeared elsewhere” [1]. That may be true of their specific definition. Still, a method established by Apollonius of Perga and later re-discovered by John Wallis calculates the derivative in a manner that feels very much like Marsden and Weinstein’s approach [2].

Math historian John Suzuki, in his essay The Lost Calculus, explains how the definition of the tangent line used by Apollonius and Wallis can be described in modern terms:

Suppose we wish to find the tangent to a curve y = f(x) at the point (a, f(a)). The tangent liney = T(x) may be defined as the line resting on one side of the curve; hence weither f(x) > T(x) for all x ≠ a, or f(x) < T(x) for all x ≠ a. (This is generally true for curves that do not change concavity; if the curve does change concavity, we must restrict our attention to intervals where the concavity is either positive or negative.)

Suzuki explores the work on derivatives by René Descartes, Jan Hudde, and Isaac Barrow, all of whom approached the problem without the need for infinitesimals and limits. Barrow even proved a version of The Fundamental Theorem of Calculus [2]!

The Limitations of Calculus Unlimited

There are many things that I like about Marsden and Weinstein’s textbook. But I’m not a calculus student anymore. I’ve been out of the classroom for nearly a decade, so it’s hard for me to say how students would respond to these ideas.

Marsden and Weinstein’s definition of the derivative is verbose, and it doesn’t address the quantifier complexity introduced by the formal definition of a limit. However, it provides a geometric algorithm for finding the derivative and avoids introducing logic that feels circular.

Although their approach has its appeal, a glance through the book is enough to see how arduous the calculations are using the book’s definitions. Whatever is gained in intuition is offset by painfully boring algebraic manipulations. Not to mention that removing limits from calculus is a disservice to students who aspire to become mathematicians.

There is a lesson here, though, that shouldn’t be overlooked.

Calculus Re-re-discovered

Newton’s and Leibniz’s ideas are indisputably important, but framing calculus as the study of limits gives students an inaccurate picture of what calculus really is. So much emphasis is placed on the limit that, at least in many students’ minds, limits become synonymous with calculus itself. That’s like saying that painting is the art of using a paintbrush!

Even worse, focusing on limits hides centuries of effort that provides historical context and validates students who struggle with the modern definitions of the limit and derivative.

Exploring derivatives without limits provides the opportunity to better understand the content of calculus, not just the tools, as well as the long history behind the problems calculus solves. In doing so, you might just come away with a better appreciation of why limits are useful. I certainly did.

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[1] Marsden, Jerrold and Weinstein, Alan J. (1981) Calculus Unlimited. Benjamin/Cummings Publishing Company, Inc. , Menlo Park, CA. ISBN 0–8053–6932–5. https://resolver.caltech.edu/CaltechBOOK:1981.001

[2] Suzuki, Jeff. (2005). The Lost Calculus. Mathematics Magazine, vol. 78, (2005), pp. 339–353. https://www.maa.org/programs/maa-awards/writing-awards/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits

The diagrams and figures in this article were created with Canva and Geogebra.

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