How are differentials understood in economics? Conceptions identified in a textbook analysis

Abstract

Differentials are commonly used in economics. However—similarly to other concepts—the way differentials are taught in mathematics courses for economics students might not fit to how they are used in subjects of the students’ major discipline. We therefore investigated by means of a textbook analysis how differentials are used and understood in microeconomics courses, and compared this with the way they are conveyed in mathematics for economics students. This analysis especially shows discrepancies between how differentials are introduced in mathematics and common ways of thinking about differentials in microeconomics, which can hinder students in gaining a holistic picture of the concept. Based on this analysis, we propose consequences for the teaching of differentials in mathematics courses for economics students.

1 Introduction

Differentials played an important role in the development of calculus during the 17th and 18th centuries (Ely, 2021). However, they disappeared from calculus at the end of the 19th century because mathematicians had not managed to make calculus based on differentials rigorous at that time (Martínez-Torregrosa et al., 2006; Ely, 2021). Instead, Weierstraß developed a different approach to calculus based on the limit concept. This approach prevailed as it was considered logically secure (Harnik, 1986). Accordingly, differentials have also been abandoned from most basic calculus and analysis courses (Thompson, 1914). Nevertheless, differentials are still commonly used in the STEM disciplines such as physics or engineering (Dray & Manogue, 2010; Alpers, 2017). Concerning physics, for instance, which had been closely interwoven with mathematics at the time when differentials were invented (Galili, 2018), there are multiple reasons why differentials have continued to be used. First, they are very practical and can simplify calculations much (Dray & Manogue, 2010). But more importantly, differentials also have their own contextual meanings in physics—mostly as infinitesimally small quantities (López-Gay et al., 2001). But it is not only the STEM disciplines that use differentials. They are also used in economics, and economics students may already be confronted with them in their basic or intermediate courses such as microeconomics (Varian, 2014). Therefore, it is important that economics students acquire an understanding of differentials.

However, gaining an understanding of differentials is no trivial matter, as several empirical studies have shown (Tall, 1980; Orton, 1983; Artigue et al., 1990). In addition, there are indications that the way differentials are introduced and treated in mathematics courses differs from how they are used and understood in other disciplines. As already said, physicists typically think of differentials as infinitesimally small quantities (López-Gay et al., 2001)—for example, as an infinitesimally small displacement or as an area or volume element (Von Korff & Sanjay Rebello, 2014; Griffiths, 2017). In mathematics courses, however, differentials are often introduced via linear approximation (Dray & Manogue, 2010). Dray & Manogue (2010) additionally highlighted that mathematicians usually introduce differentials of functions, while physicists often make use of differentials in equations describing relationships between physical quantities. Such discrepancies between how concepts are introduced in mathematics and how they are used in other disciplines might lead to gaps for students enrolled in study programmes of these disciplines (Christensen, 2008). It is thus important to investigate how differentials are used and understood in the students’ major discipline in order to help them bridge such gaps.

In the study we present here, we investigated this issue for economics, because differentials have a high relevance in core subjects of economics such as microeconomics (Voßkamp et al., in prep). Since there are multiple ways to understand differentials (Ely, 2021), which we will call ‘conceptions’ from now on, our specific research question was

Which conceptions of differentials are relied upon in microeconomics courses, and which are conveyed in mathematics courses for economics students?

We explored this question by means of a textbook analysis. Based on this analysis, we then present suggestions for the teaching of differentials in mathematics courses for economics students that bridge to how differentials are understood in microeconomics. Furthermore, our results broaden the findings about how calculus is used in other disciplines in general, which is important in order to build an alignment between this usage and the way calculus is taught in mathematics service courses.

2 Literature review

2.1 Understanding of mathematical concepts in the context of other disciplines

Since we aimed at investigating how differentials are understood in economics, we first present important findings on the understanding of mathematical concepts in the context of other disciplines. As there are only few findings related to economics thus far, we also present examples from other disciplines.

Understanding of mathematical concepts in other disciplines emerges to a large extent from contextual interpretations of the concepts. A prototypical example is the interpretation of the derivative in dynamical motion contexts in physics as speed (Zandieh, 2000; Hitier & González-Martín, 2022). Another example from electrical engineering is the interpretation of the coefficients |${c}_k$| of a Fourier expansion |$f(t)={\sum}_{k=-\infty}^{\infty }{c}_k{e}^{j{\omega}_kt}$| as the amplitude of the components that comprise a periodic signal (Rønning, 2021). Such contextual interpretations are essential for giving meaning to abstract mathematical concepts in the corresponding disciplines. Furthermore, they are also used to construct arguments justifying mathematical techniques used in other disciplines. Hochmuth & Peters (2021), for example, illustrated this for the concept of complex numbers, and its usage in signal theory to solve tasks on signal transformations. An important example of a contextual interpretation in economics is the interpretation of the derivative |${f}^{\prime }(x)$| as the change in |$f$| when increasing |$x$| by one unit—for example, as the additional cost of the next unit in the case of the marginal cost |${C}^{\prime }(x)$| (Feudel & Biehler, 2021). This interpretation is very useful in economics, as it gives a meaning to |${C}^{\prime }(x)$| and other derivatives that also makes sense in discrete settings. Furthermore, reasoning based on this interpretation as an amount of change is less demanding than reasoning based on the concept of rate (Carlson et al., 2002).

However, sometimes there are discrepancies between the way mathematical concepts are understood and taught in mathematics and the way they are understood in other disciplines. Alpers (2018), for instance, showed this for the concept of continuity by analyzing a textbook of technical mechanics. In this book, continuous functions are presented as functions without gaps or holes. This is not consistent with the mathematical definition of continuity, according to which a function is continuous if it is continuous at every point of its domain. A similar discrepancy was also found by Hochmuth et al. (2014) for the concept of the delta distribution and its usage in signal theory. Feudel & Biehler (2021) identified such discrepancies for an example that is relevant for economics: the derivative |${f}^{\prime }(x)$| and its common economic interpretation as the change in |$f$| when increasing |$x$| by one unit, e.g. as the additional cost of the next unit in the case of a cost function |$C$|⁠. While the derivative |${C}^{\prime }(x)$| is a rate of change, its interpretation as the additional cost of the next unit suggests it represents an amount of change. Furthermore, |${C}^{\prime }(x)$| and the cost of the next unit differ numerically, and this difference cannot be neglected for the prototypical functions students know from school. Feudel & Biehler (2022) then showed in an interview study that economics students struggled to make a connection between the mathematical concept of the derivative and this common economic interpretation and to explain why this interpretation is useful and legitimate in economics.

In the research presented here, we investigated how differentials—a concept of high relevance in economics (Voßkamp et al., in prep.)—are understood in economics, and whether there are also such discrepancies to the understanding conveyed in mathematics courses for economics students.

2.2 Literature review on the teaching and learning of differentials

Since our study focuses on the understanding of differentials, we reviewed literature on possible conceptions of differentials that are typically conveyed in (general) calculus/analysis courses, and on conceptions of differentials learners hold. These then formed the first basis for our analysis on how differentials are understood in economics and how they are conveyed in mathematics courses for economics students.

2.2.1 Conceptions of differentials typically conveyed in calculus courses

In basic calculus/analysis courses, differentials are often not treated as a concept in its own right. For example, in common textbooks for first-semester mathematics students in Germany, differentials are just introduced as a notation for the limit of the difference quotient (Heuser, 2003; Forster, 2016; Fritzsche, 2020). Although Heuser and Fritzsche also briefly mention Leibniz’s original idea of considering differentials as infinitesimally small quantities, they do not base their explanations related to the introduced calculus concepts on this idea. Hence, the differentials remain just formal symbols. Later, in the context of integration, differentials occur again, but the authors only use them as symbols to be manipulated in the substitution technique of integration.

Other textbooks introduce differentials as a concept in its own right in the context of local linear approximation. In this case, they are often defined via |$df:= {f}^{\prime }(x) dx$| with |$dx=\Delta x$| being a variable and |$df$| being the increase on the tangent line. Alternatively, they are directly defined as the linear mapping |$L$| that is the best local linear approximation of |$f$| in a neighbourhood of |$x$|⁠, i.e., |$\underset{h\to 0}{\lim}\frac{f\left(x+h\right)-f(x)-L(h)}{h}=0$| (Königsberger, 2004; Walter, 2004).

In the abovementioned approaches, differentials only play a minor role despite their importance in the STEM disciplines (López-Gay et al., 2001). Only a few attempts of teaching calculus that make differentials fundamental are documented in the mathematics education literature. One of these was put forward by Thompson et al. (2019). In this approach, differentials are introduced as variables in a linear relationship first. Later, the concept of integration is established via accumulation using this conception of differential. Another suggestion was put forward by Ely (2017). He introduced differentials as infinitesimally small numbers using the idea of an infinite microscope. He then developed a whole calculus course based on differentials defined as such infinitesimally small numbers.

Hence, overall, differentials are typically introduced in calculus as mere symbols for the derivative or the integration variable. Sometimes, they are also defined as objects that carry a semantic meaning. In this case, they are introduced in the context of local linear approximation as variables or as a linear mapping. The latter conception can later connect to the conception of differential form that is usually covered in differential geometry (Agricola & Friedrich, 2002). Rarely are differentials introduced as infinitesimally small numbers or quantities. However, the idea might be mentioned briefly.

2.2.2 Students’ conceptions of differentials

Since differentials usually appear at some position in calculus—even if only as a notation, it is not surprising that students have various conceptions of differentials. Tall (1980), for example, listed the following conceptions of the differential |$dy$| that he found when asking students for the meaning of the differentials in the derivative definition |$\frac{dy}{dx}=\underset{\delta x\to 0}{\lim}\frac{\delta y}{\delta x}$|⁠:

• |$dy$| has no meaning in itself

• |$dy$| means “with respect to |$y$|” (as an integration variable)

• |$dy$| is an infinitesimal increase in |$y$|

• |$dy$| is the limit of |$\Delta y$| as |$y$| gets small

• |$dy$| is a very small increase or ‘the smallest possible’ increase in |$y$|

In the first two conceptions, the differential does not have a real semantic meaning, but is a mere notation. The next two conceptions are similar to the idea of an infinitesimally small number. In the last one, |$dy$| is a small but finite increment. Similar conceptions were also found by Orton (1983) and Artigue et al. (1990).

Two more recent studies have illuminated students’ conceptions of differentials further. Hu & Rebello (2013) investigated how students from a calculus-based physics course reasoned about differentials when working on physics problems. They discovered the following four conceptions: 1) differential as a small amount, 2) differential as a point, 3) differentials as a command to take the derivative and 4) differential as an integration variable. While the latter two conceptions represent processes to be carried out, the first two also carry a semantic meaning. In the first one, the meaning is related to Leibniz’s idea of infinitesimally small quantities. However, the students in the study spoke only of ‘small’ or ‘tiny’ quantities (for instance, tiny segments of charge). Hence, this conception aligns with the ‘small increase conception’ from Tall. In the conception ‘differential as a point’, the students talked about a change as if it were just concentrated in one point. This fits to the so-called ‘collapse metaphor of limit’ in the mathematics education literature (Oehrtman, 2009), according to which dimensions get lost after taking the limit. Nilsen & Knutsen (2023), who investigated the meaning engineering students assigned to the differentials in the fundamental theorem of calculus, also found similar conceptions. In their study, some students also referred explicitly to the idea of infinitesimally small quantities. Overall, these newer studies basically show that students associate the conceptions of differentials identified by Tall (1980) also in physical or geometric contexts.

An idea students rarely seem to associate with differentials in context-oriented tasks is local linearization. López-Gay et al. (2015) investigated how high school and university students justified the usage of differentials in physics problems and how they interpreted them in physical expressions. One of their tasks was to state the meaning of a differential |$dN$| in the equation |$dN=-\lambda \cdotp N\cdotp dt$| describing radioactive decay. In the answers obtained, only one of the 70 participating university students referred to linearization, asserting that |$dN$| is a very small |$\Delta N$| so that ‘any other magnitude is thought to be constant’ (p. 607). Instead, 64% of them referred to differentials as small or infinitesimally small quantities—a conception that is commonly used in physics courses (p. 606). This suggests that the students were not aware of the local linearity assumptions behind the equation |$dN=-\lambda \cdotp N\cdotp dt$|⁠. In addition, since a conception of differentials as infinitesimally small quantities is usually not taught in calculus, this furthermore indicates a disconnect between the way differentials are typically introduced in calculus and how they are understood and used in the students’ major discipline.

Overall, the literature review on students’ understanding of differentials points out conceptions that are used by students commonly: differential as a formal symbol or as a small (finite or infinitesimally small) quantity. These conceptions especially do not coincide with the conception of differential as a variable in a linear approximation or as the approximating linear mapping |$L$|—the way differentials are defined in calculus if they are covered as an individual concept. In our study, we now tried to identify conceptions of differentials relied upon in microeconomics, and to find out whether there are also such discrepancies to how differentials are conveyed in mathematics for economics students.

3 Theoretical framing

Since we wanted to find out how differentials are understood in economics—or more precisely which understanding is relied upon when working with differentials in microeconomics courses at university, we used for our research the constructs way of understanding and way of thinking by Harel (2008). Harel defined way of understanding as follows (p. 490):

A person’s statements and actions may signify cognitive products of a mental act carried out by the person. Such a product is the person’s way of understanding associated with that mental act.

Such mental acts can be interpreting, explaining, modeling, connecting, generalizing, symbolizing, etc. From this perspective, the conceptions of differentials presented in Section 2.2 like the ones revealed by Tall (1980) can also be considered as possible ways of understanding differentials, as they either represent textbooks authors’ explanations related to the concept or students’ interpretations of it.

Based on the construct way of understanding, Harel then defined way of thinking as follows (p. 490):

Repeated observations of one’s way of understanding may reveal that they share a common cognitive characteristic. Such a characteristic is referred to as a way of thinking associated with that mental act.

In the research presented here, we aimed to find out about possible ways of understanding differentials and common ways of thinking about differentials relied upon in microeconomics courses. For this, we analyzed microeconomics textbooks used in such courses in order to reconstruct the authors’ ways of understanding and thinking about differentials, whom we considered as experts in the subject and as expert teachers of corresponding courses. Of course, we face the problem that we could not look into the authors’ minds and could only analyze how they communicate about differentials. However, we assume that thinking cannot be separated from communication—similar as in commognitive approaches (Sfard, 2008). In addition, analyzing the authors’ writings about differentials to reconstruct their understanding aligns with the theoretical constructs by Harel (2008), as the authors’ texts represent signifiers of the cognitive products of their mental acts involving differentials.

Finally, to find out about possible mismatches between the understanding of differentials relied upon in microeconomics and the one conveyed in mathematics courses, we additionally investigated the ways of understanding and thinking about differentials communicated in common mathematics textbooks for economics students.

4 Methodology

As already said, we carried out a textbook analysis in which we tried to reconstruct the authors’ ways of understanding differentials and common ways of thinking about differentials. Since textbooks play an important role in economics teaching at university, which has also been supported empirically (Beckenbach et al., 2016), we assume that the ways of understanding and thinking about differentials communicated in the textbooks also represent the ones relied upon in corresponding economics courses.

4.1 Selection of textbooks

We analyzed three microeconomics textbooks. The first one, ‘Intermediate microeconomics’ (Varian, 2014), is a very common textbook in many countries—including Germany (Beckenbach et al., 2016). We analyzed the international edition (in English) here. The other two microeconomics books we chose represent a dichotomy. One is the book ‘Microeconomics: Theory and Applications’ by Perloff (2021), which has—according to the author—a relatively practical focus and also includes real-world problems. The third book, ‘Microeconomic theory’—written by Wetzstein (2013), is much more theoretical. However, it still aims at undergraduate students. We chose these three rather different textbooks to get a broad picture of possible ways of understanding differentials students might be confronted with in microeconomics courses. For comparing these ways with the ones conveyed in mathematics courses, we additionally analyzed the ways of understanding and thinking about differentials communicated in three common mathematics textbooks for economics students (Hoy et al., 2001; Sydsæter et al., 2012; Jacques, 2018). Concerning Sydsæter et al., we analyzed the German translation (Sydsæter et al., 2015). We only focused on the chapters covering differential calculus of functions in one variable in the mathematics textbooks, as functions of two variables are not always covered in calculus courses for economics students.

4.2 Method of analysis

Our analysis consisted of three steps.

1st Step—Identification of the authors’ different ways of understanding differentials: For each textbook, we analyzed the main text according to the ways of understanding the author relies upon. The unit of analysis was each problem that the author approaches using differentials.¹ More precisely, we categorized for each of these units the ways of understanding differentials the author communicates. As categories for the coding, we first tried to use the conceptions of differentials that we identified in the literature review in section 2 (Tall, 1980; Hu & Rebello, 2013; Ely, 2021) that can be considered as possible ways of understanding differentials from the perspective of Harel (2008). These were:

1. Differentials as a sign for the derivative

2. Differentials as a shorthand for the limit of the difference quotient

3. Differentials as abstract symbols to be manipulated

4. Differential as a linear approximation

5. Differential as a linear mapping or as a differential form

6. Differential as an infinitesimally small quantity

7. Differential as a small, but finite, increment

We excluded differentials of higher order like |$\frac{d^2y}{d{x}^2}$| and differentials in integral expressions from the analysis, as these were always used only as a notation.

The first author then tried to analyze the textbooks using the above-mentioned categories 1–7. However, he recognized that the category system did not perfectly fit to the conceptions communicated in the textbooks: some did not occur, like the conception as a linear mapping, and further conceptions were found. Therefore, we refined the category system so that the categories would represent the conceptions of differentials actually found in the textbooks. A detailed description of this final category system will be presented in the Results section, as these categories already represent the ways of understanding differentials that we found in our analysis.

2nd Step—Identification of the authors’ common ways of thinking about differentials: According to our theoretical framework, ways of thinking are repeated observations of one’s way of understanding. Therefore, in order to identify the authors’ common ways of thinking about differentials, we coded how often the textbook authors used each of the different conceptions found. More precisely, we captured in a tabular raster each problem an author approaches using differentials and the conceptions communicated. This coding is illustrated in Fig. 1 for the beginning of the textbook by Wetzstein (2013). We entered a ‘1’ for each conception the author refers to in their explanations relating to the respective problem. We also captured whether an author refers to multiple conceptions in one problem by making multiple entries into one row. To make the coding more reliable, we coded separately first, then compared our results, and finally resolved disagreements in a discussion.

Regarding the coding of category 1 ‘Differentials as a sign for the derivative or slope’ in Fig. 1, we want to mention two additional remarks:

• 1) We only assigned this category if none of the differentials in the unit of analysis is interpreted because otherwise it would appear every time whenever any calculation using the derivative is carried out.

• 2) We did not count as a communicated contextual interpretation of differentials |$\frac{dy}{dx}$| if the author just mentions the name of an economic quantity the quotient |$\frac{dy}{dx}$| represents (like marginal cost), although this quantity carries a meaning. The reason is that this meaning is not communicated explicitly, and we aimed to investigate the ways of understanding the authors rely upon in their explanations. In addition, there are multiple ways to interpret such marginal quantities (Götze, 2010), and we do not know which one the author is relying upon in this situation.

3rd step—Comparison of the textbooks: After the analysis of the individual textbooks, we compared the results of the analyses of the different textbooks in order to

• identify common ways of thinking about differentials used in microeconomics,

• find possible mismatches to the ways of understanding/ways of thinking about differentials conveyed in the mathematics textbooks.

5 Results of the analysis

5.1 Conceptions found in the analyzed textbooks

As already said, not all conceptions of differentials that we considered as a priori categories for the coding occurred in the textbooks. The conception ‘linear mapping’ did not occur, so we dropped it in the further analysis. Instead, two further ways of understanding differentials were communicated by the authors of the textbooks: ‘Quotient of differentials interpreted as a rate of change’ and ‘Quotient of differentials with discrete interpretation’. Altogether, we identified nine conceptions that were used by the authors. These are shown in Table 1. In the first three of these, differentials only appear as quotients and are only used as a notation—either for a slope, a derivative, or the limit of a difference quotient. In conceptions 4 and 5, the differentials also appear exclusively as quotients. But this time, the quotients are interpreted in the context. And finally, we found four conceptions in which differentials are also considered as individual objects (conceptions 6–9). In the first one, they are just symbols that are manipulated according to certain rules but not interpreted. In the other three conceptions, the authors also assign a meaning to the differentials as stand-alone objects explicitly.

In the following, we explain the nine conceptions and their occurrence in more detail, and present illustrative examples for these from the textbooks.

5.1.1 Differentials as a mere sign for derivative or slope

In this conception, the differentials are just used as a sign for the derivative or slope without an explicit interpretation in the context. An example is the following calculation in the microeconomics textbook Perloff (2021) relating to the problem of how to maximize a monopolist’s profit (p. 391):

‘The monopoly chooses output|${Q}^{\ast }$| to maximize its profit by using the necessary condition that the derivative of its profit function with respect to the output equals zero:

$$ \frac{d\mathrm{\pi} \left({\mathrm{Q}}^{\ast}\right)}{dQ}=\frac{dR\left({Q}^{\ast}\right)}{dQ}-\frac{dC\left({Q}^{\ast}\right)}{dQ}=0,(11.1) $$

where|$dR/ dQ= MR$| is its marginal revenue function and|$dC/ dQ= MC$|is its marginal cost function. Thus, Equation (11.1) requires the monopoly to choose that output level|${\mathrm{Q}}^{\ast }$|such that its marginal revenue equals its marginal cost:|$MR\left({Q}^{\ast}\right)= MC\left({Q}^{\ast}\right)$|’

The differentials used in this derivation are used as a sign for the derivative. Although the author also mentions the name of the economic quantities the derivatives above represent, he does not communicate their meaning in this situation explicitly.

An example in which differentials are just used as a sign for slope without interpretation in the context is Wetzstein’s discussion of whether a monopolist should raise the price as much as possible to attain the highest profit (Wetzstein, 2013, p. 478). The answer is ‘no’ because the demand then decreases. Wetzstein discusses this with a linear inverse market demand curve |$p=a- bQ$|⁠, which describes the price a monopolist may set for a certain demand. Wetzstein then visualizes such a function and notates on the graph “Slope, |$\frac{dp}{dQ}=-b$|” without any contextual interpretation. Hence, the differentials are just used as a sign for the slope, but are not interpreted in the context.

The conception ‘Differentials as a mere sign for derivative or slope’ is typically used by the authors when solving problems with the derivative in calculations, for instance optimization problems, or when symbolizing the slope without interpreting the differentials in the context.

5.1.2 Differentials as an operator

In this conception, the differentials are used as a sign that demands the differentiation of a specific expression. An example is a solved problem in the microeconomics textbook² by Perloff (2021), in which he determines the marginal cost for the function |$C(q)=100q-4{q}^2+0.2{q}^3+450$| as follows (p. 240):

Here, the specific function to be differentiated is explicitly plugged in.

The mathematics textbook by Sydsæter et al. (2015) introduces this conception even explicitly (p. 202):

‘We can understand the symbol|$d/ dx$|as a command to differentiate the subsequent expression with respect to|$x$|.’

This conception is used in the analyzed textbooks from time to time whenever specific functions to be differentiated are first plugged in.

5.1.3 Differentials as shorthand for limits

In this conception, the differentials are used as a notation for the limit of the difference quotient. An example is the definition of the elasticity of substitution in Wetzstein (2013, p. 269):

Thereby, |$K/L$| represents the ratio of capital and labor, and |$MRTS$| denotes the marginal rate of technical substitution. The latter measures how much of a factor is needed to replace a certain amount of another factor in a production. Also here, the differentials are just used as a notation and are not interpreted in the context.

This conception is especially used in the analyzed textbooks for the definition of the derivative or when defining certain elasticities, as in the sample above.

5.1.4 Quotient of differentials interpreted as a rate of change

In this conception, the differentials also only appear as quotients, but the authors now interpret these differential quotients in the context as a rate of change. An example is the definition of the marginal rate of substitution in Wetzstein (2013, p. 42):

‘The marginal rate of substitution (⁠|${x}_2$|for|${x}_1$|⁠) is then defined as

$$ MRS\left({x}_2 for\ {x}_1\right)=-\frac{d{x}_2}{d{x}_1}{\left.\kern0em \right|}_{U=\mathrm{constant}} $$

[…] |$\mathrm{MRS}$|measures the rate at which a household is just on the margin of being willing to substitute commodity|${x}_2$|for|${x}_1$|. Stated differently,|$\mathrm{MRS}$|measures how much a household, on the margin, is willing to pay in terms of|${\mathrm{x}}_1$|in order to consume some more of|${x}_2$|.’

Wetzstein interprets here the differential quotient as a rate of change. However, he does not interpret the differentials |$ d{x}_2 $| and |$ d{x}_1 $| individually.

This conception particularly occurs in the analyzed economics textbooks when the authors define quantities that describe trading or exchange behaviour, such as the marginal rate of substitution above.

5.1.5 Quotient of differentials with discrete interpretation

Besides the interpretation of differential quotients as a rate, the authors also commonly use an interpretation of a differential quotient |$\frac{dy}{dx}$| as the change in |$y$| when |$x$| is increased by one unit. An example is the definition of marginal cost in the economics textbooks by Perloff (2021, p. 239):

‘A firm’s marginal cost |$(MC)$|is the amount by which a firm’s cost changes if it produces one more unit of output. The marginal cost is

$$ MC=\frac{dC(q)}{dq}.{}^{\prime } $$

This is a rather typical interpretation of the derivative in economics, which also works in discrete contexts. Thereby, the formulation above does not make clear whether the author actually thinks about the quotient |$\frac{dy}{dx}$| as an amount of change or just uses ‘amount of change’ as a phrase for the quotient |$\frac{\Delta y}{\Delta x}$| with |$\Delta x=1$|⁠. Varian (2014) mentions on one occasion that interpreting the marginal cost as the cost of the next unit is just a convenient way to think about marginal costs, but one should be aware that marginal costs are in fact a rate (p. 402). On the other hand, Perloff (2021) seems to think about the marginal costs as an actual amount |$\Delta C$|⁠, as he also uses the unit ‘dollar’ for them (see e.g., p. 532). Since these two approaches underlying the interpretation of |$\frac{dy}{dx}$| as a change in |$y$| when raising |$x$| by one unit were not always clearly distinguishable, we subsumed both under the conception ‘Quotient of differentials with discrete interpretation’.

This conception especially occurs in the analyzed textbooks when the authors interpret derivatives or marginal quantities that are symbolized with differential quotients in the context.

5.1.6 Differentials as symbols to be manipulated

In this conception, differentials of the form |$dx$| are also used as individual objects, but only as symbols that can be manipulated according to certain rules. An example is the derivation of the representation of the elasticity of substitution |$\sigma$| as a double-logarithmic derivative in the book Perloff (2021, p. 219):

‘By totally differentiating, we find that|$d\ln \left(K/L\right)=\frac{d\left(K/L\right)}{K/L}$|and|$d\ln \left(| MRTS|\right)=\frac{d(MRTS)}{MRTS}$|, so|$\left[d\ln \left(K/L\right)/d\ln \left(\left| MRTS\right|\right)\right]=\left[d\left(K/L\right)/d(MRTS)\right]\left[ MRTS/\left(K/L\right)\right]=\sigma$|.’

The differentials are manipulated here as individual objects, but without an interpretation in the situation.

This conception occurs in the analyzed textbooks in derivations connecting elasticities and derivatives of logarithmized functions as in the example above, or in calculations involving the total differential of a function without any interpretation.

5.1.7 Differentials as variables in a linear approximation

In this conception, differentials are considered as stand-alone objects and are also given a meaning. It is, for example, used in the mathematics textbook by Sydsæter et al. (2015) to introduce differentials as legitimate objects. The authors first define the differential |$dx$| as an increment in |$x$|⁠, and then define |$dy$| via |$dy={f}^{\prime }(x) dx$| (p. 269). Afterwards, they explain that |$dy$| represents the change in |$y$| when |$x$| is changed by |$dx$| if both variables were related by the constant rate of change |${f}^{\prime }(x)$|⁠. Finally, they illustrate how |$dx$|⁠, |$dy$|⁠, and |$\Delta y$| are related (see Fig. 2). Hence, |$dx$| and |$dy$| represent variables in the (best) local linear approximation of |$f$| in a neighbourhood of |$x$| using the tangent through |$\left(x,f(x)\right)$|⁠.

This conception of differentials is the one used in the analyzed mathematics textbooks for economic students to introduce differentials as legitimate objects (Hoy et al., 2001; Sydsæter et al., 2015). Regarding the analyzed economics textbooks, this conception only occurs in the textbook by Wetzstein (2013)—in the definition of the total differential in the mathematical appendix (p. 1031).

5.1.8 Differentials as infinitesimally small quantities

This conception of differentials, which is commonly used in physics or engineering (see section 2), also occurs in two of the analyzed economics textbooks (Varian, 2014; Perloff, 2021). For example, Varian uses it when introducing the marginal rate of substitution |$d{x}_2/d{x}_1$|⁠, which he describes as the willingness to exchange good 2 for good 1. For this, he argues as follows (p. 48):

“Now we think of|$\varDelta{x}_1$|as being a very small change—a marginal change. When we imagine|$\varDelta{x}_1$|becoming infinitesimally small, we choose the notation|$d{x}_1$|. Then the rate|$d{x}_2/d{x}_1$|measures the marginal rate of substitution of good 2 for good 1, […]”.

The author explicitly assigns here for the differential |$d{x}_1$| the meaning of an infinitesimally small quantity. However, he does not further define what the term ‘infinitesimally small’ means for him.

This conception does not occur in the mathematics textbooks for economics students—probably because the authors do not consider this conception as rigorous enough, as typical for mathematics textbooks (see Section 2).

5.1.9 Differential as a finite increment

The authors of the analyzed economics textbooks often communicate about differentials as if these represent finite increments. An example is Varian (2014), when he explains the meaning of the marginal utility—defined as the limit of the difference quotient (p.65):

He then argues:

‘This definition implies that to calculate the change in the utility associated with a small change in consumption of good 1, we can just multiply the change in consumption by the marginal utility of the good:

$$ dU=M{U}_1d{x}_1.\text{'} $$

Although he directly refers to the limit definition of marginal utility, he then interprets the occurring differential |$d{x}_1$| just as ‘a small change’, and not as an ‘infinitesimally small change’. Hence, in this situation, he refers to the differential |$d{x}_1$| as if it were a finite increment.

This kind of understanding of a differential as ‘a small change’ or ‘slight change’—sometimes also verbalized as ‘a marginal change’ (Varian, 2014, p. 48)—is communicated in the analyzed economics textbooks in many situations. Sometimes the authors even speak simply just of ‘a change’ in the sense of any change. For instance, Wetzstein (2013) denotes the differential |$\tau =d{p}_1$| in a problem on whether raising a tax on gasoline is useful just with “the price change |$\tau =d{p}_1$|” (p. 141). He even allows the differentials |$d{x}_i$| of the independent variables to be zero (p. 44).

The conception of differential as a finite increment especially occurs in the analyzed economics textbooks in arguments involving the total differential.

5.2 Common ways of thinking about differentials among the authors

As explained in Section 4.2, we captured for each problem in the main text of the books that involves differentials the conceptions the authors communicate when explaining the problem or its solution. Table 2 now shows how often each author uses the different conceptions of differentials found in our analysis. The first three columns represent the economics textbooks, the last three the mathematics textbooks. The numbers in the first row are the number of problems in which differentials occur within the book. The rows usually add up to more than 100% because authors sometimes relied upon several conceptions during the same problem. For example, an author could plug in a specific expression to differentiate using the operator conception of differentials, and later interpret the obtained result in the context. An exception is the category ‘Differentials as a mere sign for derivative or slope’ that we only assigned if the author does not interpret any of the differentials when tackling the respective problem, because it would otherwise appear almost every time (see Section 4).

Table 2 first reveals ways of thinking about differentials that students might be confronted with commonly in microeconomics courses—indicated by the bold entries (proportions over 10%) in the first three columns. These are as follows:

• Differentials as a notation for the derivative/slope or as an operator,

• Quotient of differentials interpreted as a rate of change,

• Quotient of differentials with a discrete interpretation,

• Differential as a finite (usually small) increment.

A connection of differentials to the limit concept and the conception of variables in a linear approximation are not common in the analyzed economics textbooks. Furthermore, thinking about differentials as infinitesimally small quantities also does not seem to be prevalent in economics—at least according to the authors’ writings in the analyzed microeconomics textbooks. Instead, differentials are interpreted rather as (small) finite increments. Finally, Table 2 suggests that if differentials are used as individual objects apart from quotients in economics, they are mostly also interpreted somehow and rarely used just as formal symbols without meaning.

Nevertheless, Table 2 also indicates differences between the authors’ common ways of thinking about differentials. Perloff (2021), for instance, mostly interprets quotients of differentials |$\frac{dy}{dx}$| in a discrete manner as the change |$\Delta y$| if |$\Delta x=1$|⁠, while the other two authors more often state interpretations as a rate of change, for example as change per unit. A reason could be that Varian (2014) and Wetzstein (2013) might want to emphasize the rate character of differential quotients (Varian even accentuates this explicitly, p. 402). Furthermore, Table 2 indicates that Perloff draws upon differentials as individual objects much less often than Varian and Wetzstein. In particular, although he occasionally interprets differentials in quotients as individual objects, he only uses them once as individual objects in a calculation (p. 219). Varian and Wetzstein, on the contrary, do so regularly, and then usually interpret the differentials as finite increments. Especially Varian (2014) draws upon this kind of thinking about differentials in many derivations. Hence, overall, Table 2 suggests that the frequency of occurrence of different conceptions of differentials in microeconomics courses might depend to a great extent on the thinking the teacher prefers.

5.2.1 Comparison between the mathematics and the economics textbooks

A comparison of the analyzed (micro)economics textbooks (first three columns in Table 2) with the mathematics textbooks shows the following commonalities:

• Differentials in quotients are often just used as a notation for the derivative without interpreting them.

• Differentials are rarely connected to the limit of difference quotients—mostly just in the definition of the derivative or of elasticity.

• Differentials appearing as stand-alone objects are rarely used just as symbols in formal calculations without any meaning (the integration technique of substitution that commonly relates to this conception does not play a role in economics), but are usually also interpreted in the context.

• The authors rarely communicate a conception of differentials as infinitesimally small quantities when interpreting them—in the mathematics textbooks even never.

However, there were also some differences between the mathematics and the economics textbooks:

• In the economics textbooks, the idea of linear approximation appeared only once.

• The authors of the economics textbooks often interpreted differentials as finite increments of quantities—often termed ‘small’, ‘slight’, or ‘marginal’ changes.

These differences indicate an important discrepancy between how differentials are conveyed in mathematics courses for economics students and the way they are thought about in economics. The analyzed mathematics textbooks that introduced differentials as legitimate objects defined them as variables in a linear approximation. However, this idea is not referred to by the authors of the analyzed economics textbooks when working with differentials in economic contexts. Instead, they tend to communicate about differentials as if a (small) change |$dx$| multiplied by a rate of change |$\frac{dy}{dx}$| would yield the exact change in |$y$|⁠. This might lead to a perceived disconnect by students that might confuse them, and hinder them from gaining a holistic picture of the concept of differentials.

6 Summary, discussion and outlook

6.1 Summary and discussion of important results

Our research showed important conceptions of differentials that students might be confronted with in basic economics subjects like microeconomics. It especially showed the following common ways of thinking about differentials that are prevalent in microeconomics (see Table 2):

• Differentials as a sign for the derivative and slope,

• Differential quotients interpreted as a rate of change or discretely as an amount of change when raising the independent variable by one unit,

• Differential as a finite increment.

The conception of differentials as infinitesimally small quantities just seems to play a subordinate role in economics. This is different from engineering or physics, in which this conception is a common way of thinking about differentials (López-Gay et al., 2015), for example as an infinitesimally small displacement (Griffiths, 2017). A reason might be that the independent variable in economics often represents a discrete good (including money). This is different from (macroscopic) physics, in which the independent variable is often considered as a continuum, e.g., time (Dummett, 2000). And even if a good considered in economics is assumed as divisible infinitely often, one is usually practically only interested in evaluating the effect of a small yet finite change in this good.

Furthermore, the changes considered in economics are usually very tiny compared to the total amounts considered. An example is mentioned in Reiß (2007), who argues—in the context of utility functions—that if water is measured in |${m}^3$|⁠, a change of a cubic millimeter is really tiny. Hence, from an economist’s perspective, it is appropriate to consider such a tiny change as infinitesimally small in the context and use the concept of differential to model such a tiny (but finite) change. And vice versa: It is also reasonable to interpret differentials occurring in derivations as small but finite increments in economics.

Finally, in economics, errors resulting from calculations within a given model can often be considered as negligible. A reason is that errors resulting from discrepancies between reality and assumptions made in economic models, for instance about the behaviour of rational agents (McMahon, 2015), are usually more severe than small errors that arise in calculations involving approximations. This is a further argument for why an interpretation of differentials as small but finite increments is practical and reasonable in economics.

Lastly, we want to highlight again the discrepancy between common ways of thinking about differentials that we identified in the analyzed microeconomics textbooks, and the ways in which differentials are defined in the analyzed mathematics textbooks. The latter define them as variables in a linear approximation, while this idea is not apparent in the microeconomics textbooks—with one exception (see Section 5.2). Similar discrepancies have also been found for other concepts and disciplines (Hochmuth et al., 2014; López-Gay et al., 2015; Alpers, 2018; Feudel & Biehler, 2021). This might lead to gaps and ruptures in students’ learning. One pathway for addressing this problem would be to try bridging such discrepancies in students’ mathematics courses.

6.2 Possible consequences for teaching

We would like to propose one possibility for introducing differentials in mathematics courses for economics students, which builds a bridge from the way differentials are currently taught in mathematics for economics students to the way differentials are commonly thought about in (micro)economics if they are considered as stand-alone objects—namely as small finite increments.

The proposal is based on the idea of ‘local straightness’ by Tall (1991). According to this idea, every differentiable function can be assumed as locally straight. This can, for example, be made visible by zooming into the graph of the function with a so-called function microscope (Greefrath et al., 2016). After zooming in at a point |${x}_0$| sufficiently enough, one finds a neighbourhood of |${x}_0$| such that the graph of the function is a straight line, and thus has a constant rate of change. One could now introduce |$dx$| as an increment that is sufficiently small such that the function |$f$| is practically linear in |$\left[{x}_0- dx,{x}_0+ dx\right]$|⁠, i.e., linear within the accuracy of the units used. Hence, |$dx$| indicates an increment that is so small that |$f$| has a constant rate of change |$m$| between |${x}_0- dx$| and |${x}_0+ dx$|⁠, unlike |$\Delta x$| which can represent any macroscopic increment. One can also use the term marginal change or marginal unit—a term common in economics—to emphasize that the increment is sufficiently small. If |$dx$| represents such a small change, one gets the relationship |$dy=m\cdotp dx$|⁠. The derivative can then be introduced as the ratio of such small changes |$dy$| and |$dx$| without the limit concept. It then represents per definition the rate of change |$m$| of the function |$f$| ‘around’ |${x}_0$|⁠.This way of introducing differentials and the derivative as a quotient of such differentials aligns with the following common ways of thinking about differentials in economics that we identified in our textbook analysis:

• Differential as a finite (small) increment,

• Quotient of differentials as a rate of change,

• Quotient of differentials |$\frac{dy}{dx}$| as a change in |$y$| when increasing |$x$| by one unit if one unit is small enough in the context such that the function is practically linear within |$\left[x;x+1\right]$|⁠.

We do not see the necessity for introducing the concept of an infinitesimally small number in mathematics for economics students, as the numerical error between the derivative defined as the limit of the difference quotient and the quotient |$\frac{dy}{dx}$| defined in the way above can be assumed as negligible in economic contexts—at least within the economic theory.

Our proposal of introducing differentials also fits to a way of thinking about differentials that appears in physics teaching (López-Gay et al., 2015, p. 598)—the authors call it ‘differentials as an infinitesimal approximation’. For physics, they criticized this approach, because expressions like |$dI=-\alpha \cdotp I\cdotp dx$| (where |$dx$| is the thickness of a plane, and |$dI$| the intensity of a plane wave when crossing the material) are never exact if |$dx$| and |$dI$| were finite increments (López-Gay et al., 2001). For economics, on the contrary, this argument does in our opinion not count, because the changes considered in economics are really tiny compared to the amounts |$x$| at which the effect of the changes are investigated, and numerical errors induced by the approximation are negligible compared to errors that occur due to discrepancies between assumptions made in economic models and reality.

Therefore, introducing a differential |$dx$| as an increment that is small enough such that the function is practically linear in |$\Big[{x}_0- dx$|⁠;|${x}_0+ dx\Big]$| seems in our view a worthwhile and practical approach to bridge between the way differentials are currently introduced in mathematics for economics students and how differentials are understood and thought about in economics.

6.3 Outlook for further research

Finally, we want to present important limitations of our study that offer fertile avenues for further research.

First, we only analyzed some textbooks in detail. Although we tried to capture a large variation regarding the conceptions of differentials that might be used in economics—due to the choice of these three rather different textbooks—our results still depend on the books chosen. It is therefore important to investigate the extent to which our results are generalizable. Second, we assumed that the textbooks determine much of what is actually taught in the students’ courses. While there exists some empirical research supporting this assumption (Beckenbach et al., 2016; González-Martín, 2021), there may be differences in some details. But most important, we could not look into the authors’ minds and could only analyze their understanding and thinking about differentials based on what they communicated in their textbooks. Therefore, interviewing economists directly about their understanding and thinking about differentials might yield more reliable results. Nevertheless, we believe our analysis provides a good starting point for investigating how the concept of differential, which is of huge relevance in economics, is understood in this discipline.

Funding

There was no external funding, but we wish to thank the library of the Humboldt-University of Berlin for covering the open access fee for this article.

Conflicts of interest

There is no conflict of interest.

Footnotes

References

Dr. Frank Feudel

• Researcher in Mathematics Education

• Working at the Humboldt-University of Berlin in Germany since 2017

• Member of the Centre for University Mathematics Education in Germany (khdm)

• Co-Leader of the International Network of Educational Research on Mathematics in Economics (INERME) since 2024

Prof. Dr. rer. nat. Thomas Skill

• Professor for Financial Mathematics and Statistics

• Researcher in Mathematics and Statistics Education

• Working at Bochum University of Applied Sciences since 2013

• Head of the specialist group “Teaching and Learning of Mathematics in Tertiary Education” of the German Mathematician Society DMV since 2021