Isaac Newton (1642-1727) invented the differentials, calling them *evanescent quantities. *Since then, the world has been wondering what these are. Just to be sure, Newton wrote his Principia (1687) by using the methods of Euclidean geometry, so that his results could be accepted in the standard of his day (context of reconstruction and presentation), and so that his results were not lost in a discussion about the new method of these differentials (context of discovery). However, this only increased the enigma. What can these quantities be, that are so efficient for science, and that actually disappear when mathematically interesting ?

Gottfried Leibniz (1646-1716) gave these infinitesimals their common labels d*y* and d*x*, and thus they became familiar as household names in academic circles, but this didn’t reduce their mystery.

Charles Dodgson (1832-1898) as Lewis Carroll had great fun with the Cheshire Cat, who disappears but leaves its grin.

Abraham Robinson (1918-1974) presented an interpretation called “non-standard analysis“. Many people think that he clinched it, but when I start reading then my intuition warns me that this is making things more difficult. (Perhaps I should read more though.)

In 2007, I developed an algebraic approach to the derivative. This was in the book “*A Logic of Exceptions*” (ALOE), later also included in “*Elegance with Substance*” (EWS) (2009, 2015), and a bit later there was a “proof of concept” in “*Conquest of the Plane*” (COTP) (2011). The pdfs are online, and a recent overview article is here. A recent supplement is the discussion on continuity.

In this new algebraic approach there wasn’t a role for differentials, yet. The notation d*y* / d*x* = *f *‘[*x*] for *y *= *f *[*x*] can be used to link up to the literature, but up to now there was no meaning attached to the symbolism. In my perception this was (a bit of) a pity since the notation with differentials can be useful on occasion, see the example below.

Last month, reading Joop van Dormolen (1970) on the *didactics* of derivatives and the differential calculus – in a book for teachers Wansink (1970) volume III – I was struck by his admonition (p213) that d*y* / d*x* really is a quotient of two differentials, and that a teacher should avoid identifying it as a *single symbol* and as the *definition of the derivative*. However, when he proceeded, I was disappointed, since his treatment didn’t give the clarity that I looked for. In fact, his treatment is quite in line with that of Murray Spiegel (1962), “*Advanced calculus* (Metric edition)”, Schaum’s outline series, see below. (But Van Dormolen very usefully discusses the didactic questions, that Spiegel doesn’t look into.)

Thus, I developed an interpretation of my own. In my impression this finally gives the clarity that people have been looking for starting with Newton. At least: I am satisfied, and you may check whether you are too.

I don’t want to repeat myself too much, and thus I assume that you read up on the algebraic approach to the derivative in case of questions. (A good place to start is the recent overview.)

##### Ray through an origin

Let us first consider a ray through the origin, with horizontal axis *x* and vertical axis *y. *The ray makes an angle α with the horizontal axis. The ray can be represented by a function as* y = f *[*x*] = *s x, *with the slope *s *= tan[α]. Observe that there is no constant term (*c* = 0).

The quotient *y* / *x *is defined everywhere, with the outcome *s, *except at the point *x *= 0, where we get an expression 0 / 0. This is quite curious. We tend to regard *y */ *x *as the slope (there is no constant term), and at *x *= 0 the line has that slope too, but we seem unable to say so.

There are at least three responses:

(i) Standard mathematics then takes off, with *limits* and *continuity*.

(ii) A quick fix might be to try to define a separate function to find the slope of a ray, but we can wonder whether this is all nice and proper, since we can only state the value *s *at 0 when we have solved the value elsewhere. If we substitute *y *when it isn’t a ray, or example *y *= *x*², then we get a curious construction, and thus the definition isn’t quite complete since there ought to be a test on being a ray.

(iii) The algebraic approach uses the following definition of the *dynamic quotient*:

*y* // *x* ≡ { *y* /* x*, unless *x* is a variable and then: assume *x* ≠ 0, simplify the expression* y* / *x*, declare the result valid also for the domain extension *x* = 0 }

Thus in this case we can use *y* // *x = **s x *// *x *= *s, *and this slope also holds for the value *x *= 0, since this has now been included in the domain too.

##### In a nutshell for d*y* / d*x*

In a nutshell, we get the following situation for d*y* / d*x*:

Properties are exactly as Van Dormolen explained:

- “d
*y”*and “d*x”*are names for variables, and thus they have their own realm with their own axes.

- The definition of their relationship is d
*y*=*f*‘[*x*] d*x.*

The news is:

- The mistake in history was to write d
*y*/ d*x*instead of d*y*// d*x.*

The latter “mistake” can be understood, since the algebraic approach uses notions of set theory, domain and range, and dynamics as in computer algebra, and thus we can forgive Newton for not getting there yet.

To link up with history, we might define that the “symbol d*y* / d*x* as a whole” is a shortcut for d*y* // d*x*. This causes additional yards to develop the notion of “symbol as a whole” however. My impression is that it is better to use d*y* // d*x *unless it is so accepted that it might become pedantic. (You must only explain that the Earth isn’t flat while people don’t know that yet.)

##### Application to Spiegel 1962 gives clarity

Let us look at Spiegel (1962) p58-59, and see how above discussion can bring clarity. The key points can all be discussed with reference to his figure 4-1.

Looking at this with a critical eye, we find:

- At the point
*P,*there is actually the creation of two new sets of axes, namely, both the {Δ*x*, Δ*y*} plane and the {d*x*, d*y*} plane. - These two new planes have both rays through the origin, one with angle θ and one with angle α.
- The two planes help to define the error. An error is commonly defined from the relation “true value = estimate + error”. The true value of the angle is θ and our estimate is α.

- Thus we get absolute error Δ
*f**=**s*Δ*x*+ ε where*s*= d*y*/ d*x*. This error is a function of Δ*x,*or ε = ε[Δ*x*]. It solves as ε*=*Δ*f –**s*Δ*x*.

- The relative error is Δ
*f*/ Δ*x =*d*y*/ d*x*+*r*which solves as*r**=*Δ*f /*Δ*x – dy / dx*. This is still a function*r*[Δ*x*]. We use the quotient of the differentials instead of the true quotient of the differences.

- We better re-consider the error in terms of the dynamic quotient, replacing / by // in the above, because at
*P*we like the error to be zero. Thus in above figure we have ε = Δ*f**–**s*Δ*x**,*where*s*= d*y*// d*x*. - A source of confusion is that Spiegel suggests that d
*x*≈ Δ*x*or even d*x =*Δ*x*but this is numerically true only sometimes and conceptually there surely is no identity since these are different axes. - In the algebraic approach, Δ
*x*is set to zero to create the derivative, in particular the value of*f*‘[*x*] = tan[α] at point*P.*In this situation, Δ*x*= 0 thus clearly differs from the values of d*x*that are still available on d*x*‘s whole own axis. This explains why the creation of the differentials is useful. For, while Δ*x*is set to 0, then the differentials can take any value, including 0.

Just to be sure, the algebraic approach uses this definition:

*f* ’[*x*] = {Δ*f* // Δ*x*, then set Δ*x* = 0}

Subsequently, we define d*y* = *f *‘[*x*] d*x, *so that we can discuss the relative error *r* = Δ*f* // Δ*x – *d*y *// d*x.
*

PM. Check COTP p224 for the discussion of (relative) error, with the same notation. This present discussion still replaces the statement on differentials in COTP p155, step number 10.

##### A subsequent point w.r.t. the standard approach

Our main point thus is that the mistake in history was to write d*y* / d*x* instead of d*y* // d*x.* There arises a subsequent point of didactics. When you have real variables *x *and *z, *then these have their own axes, and you don’t put them on the same axis just because they are both reals.

See **Appendix A** for a quote from Spiegel (1962), and check that it is convoluted at times.

**Appendix B** contains a quote from p236 from Adams & Essex (2013). We can see the same confusions as in Spiegel (1962). It really is a standard approach, and convoluted.

The standard approach takes Δ*x *= d*x *and joins the axis for the variable Δ*y* with the axis for the variable d*y*, with the common idea of “a change from *y*“. The idea of this setup is that it shows the error for values of Δ*x *= d*x*.

It remains an awkward setup. It may well be true that John from Los Angeles is called Harry in New York, but when John calls his mother back home and introduces himself as “Mom, this is Harry”, then she will be confused. Eventually she can get used to this kind of phonecalls, but it remains awkward didactics to introduce students to these new concepts in this manner. (Especially when John adds: “Mom, actually I don’t exist anymore because I have been set to zero.”)

Thus, in good didactics we should drop this Δ*x *= d*x*.

Alternatively put: We might define d*y *= *f* ’[*x*] Δ*x = * {Δ*f* // Δ*x*, then set Δ*x* = 0} Δ*x. *In the latter expression Δ*x* occurs twice: both as a local and bound variable within { … } and as a global free variable outside of { … }. This is okay. In the past, mathematicians apparently thought that it might make things clearer to write d*x *for the free global variable: d*y *= *f* ’[*x*] d*x. *In a way this is okay too. But for didactics it doesn’t work. We should rather avoid an expression in which the same variable (name) is uses both locally bound and globally free.

##### Clear improvement

Remarkably, we are using 99% of the same apparatus as the standard approach, but there are clear improvements:

- There is no use of limits. All information is contained in the algebra of both the function
*f*and the dynamic quotient. See here for continuity. - There is a clear distinction between the three realms {
*x*,*y*}, {Δ*x*, Δ*y*} and {d*x*, d*y*}. - There is the new tool of the {d
*x*, d*y*} space that can be used for*analysis of variations.* - Didactically, it is better to first define the derivative in chapter 1, and then introduce the differentials in chapter 2, since the differentials aren’t needed to understand chapter 1.
- There is clarity about the error, that one doesn’t take d
*x*≈ Δ*x*but considers ε = Δ*f**–**s*Δ*x**,*where*s*has been found from the recipe*s*=*f*’[*x*] = {Δ*f*// Δ*x*, then set Δ*x*= 0}.

##### Example by Van Dormolen (1970:219)

This example assumes the total differential of the function *f*[*x, y*]:

d*f *= (∂*f* // ∂*x*) d*x* + (∂*f* // ∂*y*) d*y*

**Question**. Give the slope of the tangent in the point {3, 4} of the circle *x*²* + y² = *25.

**Answer**. The point is on the circle indeed. We write the equation as *f*[*x, y*] *= x² + y² = *25. The total differential gives 2*x *d*x *+ 2*y* d*y = *0. Thus d*y* // d*x* = – *x* // *y. *Evaluation at the point {3, 4} gives the slope – 3/4. □

PM. We might develop *y *algebraically as a function of *x *and then use the +√ rather than the -√. However, more abstractly, we can use *y *= *g *[*x*], and use d*y* = *g *‘[*x*] d*x, *so that the slope of the tangent is *g *‘[*x*] at the point {3, 4}. Subsequently we use *g *‘[*x*] = d*y* // d*x*.

PM. In the Dutch highschool programme, partial derivatives aren’t included, but when we can save time by a clear presentation, then they surely should be introduced.

##### Conclusion

The conclusion is that the algebraic approach to the derivative also settles the age-old question about the meaning of the differentials.

For texts in the past the interpretation of the differential is a mess. For the future, textbooks now have the option of above clarity.

Again, a discussion about didactics is an inspiration for better mathematics. Perhaps research mathematicians have abandoned this topic for ages, and it is only looked at by researchers on didactics.

##### Appendix A. Spiegel (1962)

Quote from Murray Spiegel (1962), “*Advanced calculus* (Metric edition)”, Schaum’s outline series, p58-59.

##### Appendix B. Adams & Essex (2013)

The following quote is from Robert A. Adams & Christopher Essex (2013), “Calculus. A Complete Course”, Pearson, p236.

- It is a pity that they use
*c*as a value of*x*rather than as an universal name for a constant (value on the*y*axis). - For them, the differential cannot be zero, while Spiegel conversely states that it is “not necessarily zero”.
- They clearly show that you can take
*f*‘[*x*] Δ*x*in in {Δ*x,*Δy} space, and that you then need a new symbol for the outcome, since Δy already has been defined differently. However, it is awkward to say: “For such an approximation, the quantity Δ*x*is traditionally denoted as*dx*(…)”. It may well be true that John from Los Angeles is called Harry in New York, … etcetera, see above.