2.3 Differentiation: Introduction to Notation
In this section, you will practise reading out mathematical symbols. This is important because your lecturers will often read mathematical statements in lectures, and you will also need to communicate about mathematics with others.
Mathematics uses two kinds of systems of communication:
- Words (for example, 'plus', 'minus', 'equals').
- Mathematical symbols (for example, [latex]+[/latex], [latex]-[/latex], [latex]=[/latex]).
The same mathematical symbols are used in every language, but you still need to know how to say them in English. Learning to read out symbols will help you understand your teachers, communicate mathematics with others, and read mathematical texts more easily.
Given a function [latex]f(x)[/latex], we denote the derivative of [latex]f(x)[/latex] by
\[f^{\prime}(x).\]
This is read out as f prime of x, or f dash of x.
The derivative [latex]f^{\prime}[/latex], is often also written as
\[\frac{df}{dx} \text{ or also as } \frac{d}{dx}f .\]
The benefit of this notation is that it explicitly reminds us that we are differentiating with respect to [latex]x[/latex]. We read these as the derivative of f with respect to x and the derivative with respect to x of f, respectively.
Another way of reading out [latex]\frac{df}{dx}[/latex] and [latex]\frac{d}{dx}f[/latex] is d f by d x and d by d x of f, respectively. Whilst these are less common, you might still encounter them in your lectures, particularly if the lecturer wants to save some time.
The process is called differentiation and the verb is differentiate. The function we obtain after differentiation is called the derivative. We can also say to take the derivative of a function.
Language Focus
When we are talking about differentiation, the verb is differentiate, not derive. The word derive is used in mathematics when we have obtained something from another thing for example, How can we derive an equation from a word problem?.
Furthermore, we shouldn't confuse these mathematical meanings of the words differentiation and differentiate, with the general meanings of these words. The general meaning of these words is to recognise what makes someone/something different, for example, What differentiates undergraduate mathematics students from postgraduate mathematics students?.
Similarly, we must distinguish between the general meaning of the word derivative, which is a substance obtained from another substance (and comes from the verb derive), for example, Cheese is a derivative of milk. You may also be interested to look up the term financial derivative.
We can proceed to differentiate this new function once more. That is, given a function [latex]f(x)[/latex], we denote the derivative of [latex]f^{\prime}(x)[/latex] by
\[f^{\prime\prime}(x).\]
This is known as the second derivative of [latex]f(x)[/latex]. The above is read out as f double prime of x, or f double dash of x.
We also often write the second derivative as,
\[\frac{d^2f}{dx^2} \text{ or alternatively as } \frac{d^2}{dx^2}f.\]
As before, the benefit of this notation is that it explicitly reminds us that we are differentiating twice with respect to [latex]x[/latex]. We read these as the second derivative of f with respect to x and the second derivative with respect to x of f, respectively.
Similarly, if we want to save time, we can also read [latex]\frac{d^2f}{dx^2}[/latex] and [latex]\frac{d^2}{dx^2}f[/latex] as d two f by d x squared and d two by d x squared of f, respectively.
Mathematics Focus
In physics and mathematical physics, particularly when the independent variable is time, i.e. [latex]y= f(t)[/latex], we often use the notation [latex]\dot y[/latex], for the derivative of [latex]y[/latex] with respect to [latex]t[/latex]. Higher order derivatives are represented with multiple dots, [latex]\ddot y[/latex], [latex]\dddot y[/latex] and so on.
[latex]\dot y[/latex] usually denotes velocity, and [latex]\ddot y[/latex] acceleration. We read these as y dot and y double dot, respectively.
Similarly, we can keep on differentiating to obtain the third derivative of [latex]f(x)[/latex], [latex]f^{\prime\prime\prime}(x)[/latex], the fourth derivative of [latex]f(x)[/latex], [latex]f^{\prime \prime \prime \prime}(x)[/latex], and so on. We of course have the analogous notations of [latex]\frac{d^3f}{dx^3}[/latex] and [latex]\frac{d^4f}{dx^4}[/latex].
Note that whilst one can read [latex]f^{\prime\prime\prime}(x)[/latex], as f triple prime of x or f triple dash of x, and [latex]\frac{d^3f}{dx^3}[/latex] as d three f by d x cubed, it is often preferred to just say the third derivative with respect to x. This is especially true for derivatives of order higher than three.
In general, we denote the [latex]n[/latex]-th derivative of [latex]f(x)[/latex] by
\[f^{(n)}(x) \text{ or }\frac{d^n f}{dx^n}.\]
Note that these are simply read out as the n-th derivative of f of x and the n-th derivative of f with respect to x, respectively.
Mathematics Focus
Note the brackets around [latex]n[/latex] in [latex]f^{(n)}(x)[/latex]. The presence of the brackets indicate differentiation, whilst the absence of brackets indicate composition of functions. That is:
- [latex]f^{(n)}(x) = \frac{d^n f}{dx^n}[/latex], whereas
- [latex]f^{n}(x) = f \circ f \circ f \circ \cdots \circ f (x)[/latex].
Furthermore, if we want to denote exponentiation, it is best to write [latex][f(x)]^n[/latex].
The expression [latex]n[/latex]-th is commonly used in mathematics. It is pronounced ennth and comes from the ordinal form of numbers such as fourth (4th), fifth (5th), and sixth (6th). It serves as a generalised ordinal number, where [latex]n[/latex] represents an arbitrary position in a sequence. Common examples of use in mathematics include:
- The [latex]n[/latex]-th derivative refers to taking the derivative [latex]n[/latex] times.
- We can now calculate the [latex]n[/latex]-th term in the sequence. (See Section 4.3)
- Find the [latex]n[/latex]-th root of a number.
Note that in sequences (see Chapter 4), when we want to distinguish between two different arbitrary positions, such as in a Cauchy sequence, we can use [latex]n[/latex]-th and [latex]m[/latex]-th.
In mathematics we also use the term zero-th (0th), as the ordinal form of 0. This is particularly common in:
- Computer science and programming, where counting often starts from 0.
- In certain mathematical contexts it is useful to call the function itself, before any differentiation, the zeroth derivative.
- In general, it is sometimes easier to start a sequence at position 0 rather than position 1.
In the next section, you will learn how to combine mathematical symbols to make complete mathematical statements, instead of studying symbols separately as you did in this section.
