2.6 Differentiation: Exploring Mathematics Through Text

The purpose of this activity is to help you think about three features of a text: its audience (who reads it), its purpose (why it was written), and its structure (how it is organised). Academic writing takes many different forms. Each form has its own conventions, purposes, and ways of organising information. Noticing these differences will help you become a stronger academic reader and writer.

Question 1: All texts have an audience, that is, a person or a group of people for whom the text is written. Look over the text and select two of the following four possible audiences for this encyclopaedia entry. Next, select the ‘Answer’ button below the options.

Possible audience 1: A lecturer in mathematics looking for some brief historical background to introduce a first lecture on differentiation to first-year undergraduate students.

Possible audience 2: A lecturer in mathematics looking for advanced techniques in the uses of differentiation for a final-year undergraduate seminar on differentiation.

Possible audience 3: A student of mathematics looking for help in how to solve problems using differentiation.

Possible audience 4: A student of the history of mathematics looking for examples of the independent development of some mathematical technique or idea.

 

Answer

The most likely audiences for this encyclopaedia extract are Audiences 1 and 4.

There are very few calculations in the extract, which makes it far less useful than, for example, lecture notes from a module or a textbook entitled Differentiation. This rules out Audiences 2 and 3, both of whom need technical content, either advanced techniques or problem-solving guidance, which this text does not provide.

Specifically:

• The lecturer in Audience 1 might draw on the brief historical information in the opening paragraph, as well as the narrative about the independent development of calculus by Newton and Leibniz and the political dimensions of this dispute. They would then cite this source in their lecture notes.

• The student in Audience 4 would similarly use the entry as a starting point for investigating the independent development of mathematical ideas. However, they are likely to need to consult further sources, as the information in this entry is relatively brief.

Question 2:  Read the whole text briefly and select which of the following is the purpose of the text. Next, select the ‘Answer’ button below the options.

1. To make an argument for the importance of differentiation in mathematics education.
2. To promote the view that Newton’s approach was preferable to Leibniz's approach in two distinct ways.
3. To present the historical development of the topic of differentiation.
4. To show that Greek mathematics achieved less compared to later mathematics in the area of differentiation.

 

Answer

The correct answer is 3.

• Option 1 is not correct: there is no mention of the educational value of differentiation.

• Option 2 is not correct: there is no suggestion that Newton's approach was superior in any way. Moreover, a large portion of the text is not specifically about Newton or Leibniz at all.

• Option 4 is not correct: the text does not make an explicit comparison between Greek mathematical achievements and modern approaches to differentiation. Greek mathematicians are mentioned, but no evaluative contrast is drawn.

Question 3: Academic texts are organised in different ways depending on their purpose and audience. Read the text again briefly and select which of the following best describes the structure of this encyclopaedia entry. Next, select the ‘Answer’ button below the options.

Answer

The correct answer is 2.

• Option 1 is not correct: the text does not group applications of differentiation. It focuses on historical development, not modern usage.

• Option 3 is not correct: the text is not a mathematical argument or proof. It is a factual, narrative account.

• Option 4 is not correct: there is no problem–solution pattern. The text does not frame differentiation as a solution to an identified problem.

A chronological structure is very common in historical encyclopaedia entries and in the history of mathematics more generally. As you read other such texts, notice how writers use time markers (such as in the seventeenth century, later, and subsequently) to signal this kind of structure.

Summarising a text is an extremely valuable way of demonstrating to yourself that you have understood it. It ensures that you have read the text carefully and it makes you check the meaning of terms that you may not fully understand, either mathematically or in terms of language.

Choose the best summary from the three options below. Be prepared to justify your selection to a partner or your tutor. For example, you might say, 'I have selected Summary X because summary Y seems irrelevant, and summary Z does not seem to emphasise the main idea of the passage'. Next, select the ‘Answer’ button below the options.

Summary 1:

Differentiation, a concept in calculus, was first developed by the ancient Egyptians. It was later formalised in the 16th century by Galileo and Descartes, who introduced the notation [latex]\frac{dy}{dx}[/latex]​. The 17th century saw contributions from Euler and Gauss, who refined the concepts of limits and continuity. Today, differentiation is primarily used in biology and literature, with limited applications in other fields.

Summary 2: 

Differentiation, a key concept in calculus, has its origins with ancient Greek mathematicians like Euclid and Archimedes. It was formalised in the 17th century by Newton and Leibniz, who introduced foundational principles and notations. The 18th and 19th centuries saw significant advancements by Lagrange, Cauchy, and Weierstrass, who formalised limits and continuity. Today, differentiation is essential in various fields, including  physics, engineering, and economics for solving problems involving rates of change.

Summary 3: 

Differentiation, a fundamental concept in calculus, was developed independently by Newton and Leibniz in the 17th century. Newton's work focused on motion, while Leibniz introduced the notation [latex]\frac{dy}{dx}[/latex]​. The 18th and 19th centuries saw contributions from Lagrange, Cauchy, and Weierstrass, who formalised the concepts of limits and continuity. During this period, the Industrial Revolution was transforming economies and societies, leading to significant technological advancements.

 

Answer

Summary 2 is the correct and relevant summary. Summary 1 is incorrect as there is no mention of Egyptian mathematics and also incorrectly mentions Galileo and Descartes. The final sentence of Summary 1 is also inaccurate both in fact and in relation to the text. Summary 3 is much closer to the text with respect to the summary of the material about Newton and Leibniz; however the final sentence, even though it is correct, it is not a good summary of the final paragraph of the text.

Now write your own summary of the text. Your summary should have the same main meaning as the correct summary above. However, it can be different in the way it is written, including the sentence structure, word choice, and how the ideas are organised.

In this activity, you are presented with five summary sentences: four that summarise each of the four paragraphs in the text, and one irrelevant summary sentence that does not fit any paragraph.
Your tasks are:

1. identify the sentence that is completely irrelevant to the text,
2. match each of the four relevant summary sentences to the four paragraphs.

After making a note of your responses, select the ‘Answer’ button below to reveal the correct combination.

 

Summary Sentence 1: A summary of the earliest steps taken in differentiation in the European context.

Summary Sentence 2: Extensions and developments: novel frameworks and greater rigour.

Summary Sentence 3: Leibniz' correspondence with Newton.

Summary Sentence 4: Two key thinkers and their distinct approach to notational matters.

Summary Sentence 5: Applications across a range of fields.

 

Answer

Summary Sentence 1 corresponds to paragraph 1. Summary Sentence 2 corresponds to paragraph 3. Summary Sentence 3 is the wholly irrelevant one. Summary Sentence 4 corresponds to paragraph 2. Summary Sentence 5 corresponds to paragraph 4.

In the previous activity, you assigned a summary sentence to each paragraph. However, paragraphs do not work independently but connect to each other and to the purpose of the text. Just like in a mathematical argument, where each logical statement is connected, as well as to the broader idea of the argument.

This activity builds on the previous one by exploring how connections are made within and between paragraphs, and how individual language elements connect to the wider purpose of the text.

Consider Paragraph 1 reproduced below with the provided annotations.

Differentiation, [a fundamental concept in calculus (1)], [enjoys a rich history dating back to ancient times (2)]. [The roots of differentiation can be traced to the ancient Greeks (3)], [particularly to mathematicians such as Euclid (c. 300 BCE) and Archimedes (c. 287–212 BCE) (4)], [who conceptualised the derivative in the sense of a tangent line (Powers, 2020; Wikipedia, 2025). (5)] [However, it wasn't until the 17th century that differentiation began to take its modern form. (6)]

Now consider Paragraph 2 and follow the procedure above:

1. divide the paragraph into phrases,
2. for each phrase, identify how it looks back to previous ideas and how it looks forwards to upcoming content,
3. note any useful language, including language that is noteworthy in itself or language that contributes to the connections between the paragraphs,
4. consider how the language relates to the overall purpose.

Answer

[In the late 17th century, two mathematicians, Sir Isaac Newton (1643–1727 CE) and Gottfried Wilhelm Leibniz (1646–1716 CE), independently developed the foundations of calculus, including differentiation.(1)] [Newton's approach was rooted in his study of motion and change, leading to the development of his famous laws of motion.(2)] [Leibniz, on the other hand, introduced the notation we use today, such as [latex]\frac{dy}{dx}[/latex]​ to express the derivative of a function [latex]y[/latex] with respect to [latex]x[/latex].(3)] [This innovation made the process of differentiation more systematic and easier to apply, and it is currently used in physics and engineering.(4)] [Moreover, Leibniz's notation is particularly useful for making the relationship between the dependent and independent variables [latex]y[/latex] and [latex]x[/latex] explicit.(5)] [Newton developed a different notation, using a dot above the function to denote its derivative, such as [latex]\dot y[/latex]​, which is commonly used in mechanics and dynamics to represent rates of change with respect to time.(6)] [Later on, Joseph-Louis Lagrange (1736–1813) introduced the prime notation, where the derivative of a function [latex]f(x)[/latex] is written as [latex]f^{\prime}(x)[/latex].(7)] [This notation is often used in mathematical analysis and is convenient for higher-order derivatives.(8)]

A possible answer is as follows.

Annotation	Comment
(1)	Looks back through the explicit echoing of the phrase In the 17th century Looks forward by naming the two central mathematicians whom sentences 2-6 will discuss further, and suggests that there may be differences between the two through the word independently.
(2)	Looks back through picking up on Newton. There is no explicit looking forward. The next sentence turns to Leibniz.
(3)	Looks back by picking up on Leibniz, indicating a clear contrast with on the other hand.
(4)	Looks back through the phrase This innovation, where further details are provided on the Leibniz notation.
(5)	Looks back to the previous sentence with Moreover providing further information on the notation.
(6)	Looks back to Newton simply by naming him. The phrase a different notation links back to the previous sentences on Leibniz's notation.
(7)	Here the text takes a new turn introducing a later period and a new mathematician. However, a look-back to notation is clear.
(8)	Looks back to the previous sentence with the phrase this notation.

In mathematics prose writing, just as in writing a mathematics proof, we will sometimes have to rewrite part of a text. This activity invites you to practise this skill.

The purpose of Paragraph 1 is to indicate that we are reading a historical text about differentiation, to make a brief comment about early contributions to the field, and to look forward to Paragraph 2.

Your task is to make any additions or modifications to Paragraph 1. Additions may include inserting a new sentence, clause or phrase at any point in the paragraph. Modifications may include deleting existing material and adding in new material. Make grammatical modifications as necessary, and consider whether your changes alter the overall purpose of the paragraph.

Example 1:

Differentiation, a fundamental concept in calculus, has a rich history dating back to ancient times. In the European tradition, the roots of differentiation can be traced to the ancient Greeks, particularly to mathematicians such as Euclid (c. 300 BCE) and Archimedes (c. 287–212 BCE), who conceptualised the derivative in the sense of a tangent line (Powers, 2020; Wikipedia, 2025). However, it wasn't until the 17th century that differentiation began to take its modern form.

No alteration of the purpose of the paragraph.

Example 2:

Differentiation, a fundamental concept in calculus, enjoys a rich history dating back to ancient times. In the European tradition, the roots of differentiation can be traced to the ancient Greeks, particularly to mathematicians such as Euclid (c. 300 BCE) and Archimedes (c. 287–212 BCE), who conceptualised  the derivative in the sense of a tangent line (Powers, 2020; Wikipedia, 2025). However, it wasn't until the 17th century that differentiation began to take its modern form. This is exemplified in the following section.

The purpose of the paragraph is altered: the paragraph is now much more focused on Greek contributions.

Consider paragraph 2 which begins 'In the late 17th century …'. We are going to study how each sentence connects to then next. To do this, we will use the ideas of given information (what the reader already knows) and new information (what is new to the reader).

• Given information: information that has already been mentioned or is known to the reader. Writers often repeat or refer to this information to make the text easier to follow.
• New information: additional information which is new and adds something extra to the reader.

Good writing usually links new information to given information to make the text flow smoothly. We will look through this paragraph and identify some connections.

In sentence 1 (S1), the given information is on the topic of differentiation. This is mentioned right at the end of S1. The new information is the remainder of the sentence: 'In the late 17th century, two mathematicians, Sir Isaac Newton (1643–1727 CE) and Gottfried Wilhelm Leibniz (1646–1716 CE), independently developed the foundations of calculus […]'.

Now, before you continue reading:

• re-read sentences 2 (S2) and 3 (S3),

• identify the given information and new information in each sentence,

• look for any linking words or phrases in S2 or S3 that help organise the information.

Next, select the ‘Answer’ button below.

 

Answer

The given information is firstly the names of the two mathematicians, Isaac Newton and Gottfried Wilhelm Leibniz, who were mentioned, in the same order, in S1. A second element of given information is that these two mathematicians introduced developments into the study of differentiation. The new information is the particular contribution of each mathematician. A linking phrase that helps to order the information is 'on the other hand'. This makes it clear to the reader that the new information about Leibniz will contrast in some way with the information about Newton.

 

Continue reading through the paragraph. For each sentence, write down what is given information (already known) and what is new information (new to the reader). Also, look for any linking words or phrases that help connect the ideas and organise the text. Next, select the ‘Answer’ button below.

 

Answer

S4: Given: 'This innovation' (i.e. Leibniz's innovation); new: the effect of innovation.

S5: Given: 'Leibniz's notation'; new: additional information of the value of Leibniz's notation; linking word or phrase: 'Moreover, …' which indicates that S5 is a development of S4.

S6: Given: 'Newton' and 'notation'; new: details on Newton's notation; linking word or phrase: 'different' in 'different notation' which indicates to the reader that details about an alternative notation are about to be discussed.

S7: Given: 'notation'; new: another individual and their contribution to the development of notation; linking word or phrase: 'Later on, …' which indicates that a further development is about to be discussed.

S8: Given: 'This notation'; new: further detail about the notation.

Locate specific words and phrases in the text with the following meanings. Next, select the ‘Answer’ buttons below.

1. Identify a single word in paragraph 1 which means something like 'understood' or 'thought about'.
   
   Answer

   conceptualised

2. Identify a two-word string in paragraph 2 which describes what mathematical ideas the approach of one of the two mathematicians was based on.
   
   Answer

   rooted in

3. Identify a four-word phrase in paragraph 3 which means something like 'get rid of any vagueness or lack of systematicity'.
   
   Answer

   eliminate ambiguities and inconsistencies

In paragraph 3, you have the read the following sentence: This notation is particularly useful for making the relationship between the dependent and independent variables [latex]y[/latex] and [latex]x[/latex] explicit.

This can be broken down in the following way:

Text	The notation	is	particularly useful	for making	the relationship	between	the dependent and independent variable	explicit.
Grammatical Analysis	subject	is	[adverb + adjective]	for making	noun phrase as object of the verb	preposition	noun phrase	resultative adjective

Consider the subpart of this sentence which reads The notation is useful for making the relationship explicit. We can make this into a schema: X + is + Adjective1 + for making + Noun + Adjective2. This schema carries the idea that something, X, has a particular quality (here useful) and that this quality can help create another quality (here explicit) in something else.

Below are some further example sentences following the above scheme.

1. The design of the table is always important for making the relationship between the data items clear.
2. This lecture was vitally important for making the connection between the two clear evident.
3. The input provided by the assessment lead was helpful in making the instructions on the exam easier to understand. 

Make sure you are familiar with the grammar of this construction. Then try to create one or two more sentences using this structure.