1.4.2 Leibniz notation

The notation that we’ve been using so far, in which the derivative of a function is denoted by , is called Lagrange notation or prime notation. (‘Lagrange’ is pronounced as a French word: ‘La-grawnge’.) It was invented by Joseph-Louis Lagrange, about a century after calculus was discovered. The term ‘prime notation’ arises from the fact that the symbol  in the notation is often called ‘prime’.

However, there’s another notation, called Leibniz notation, invented by Gottfried Wilhelm Leibniz. (Remember that ‘Leibniz’ is pronounced as a German word: ‘Libe-nits’.) You’ll need to become familiar with both notations, as they’re both used throughout this material, and throughout mathematics generally.

Each type of notation has different advantages in different situations. Generally, Lagrange notation is used when you’re thinking in terms of a variable, and a function of this variable. On the other hand, Leibniz notation is often used when you’re thinking more of the relationship between two variables. The distinction will become clearer as you become used to working with the two notations.

Joseph-Louis Lagrange

Joseph-Louis Lagrange was an Italian-French mathematician who made important contributions in many areas, including calculus, mechanics, astronomy, probability and number theory. He was appointed as a professor of mathematics at the Royal Artillery School in Turin at the age of only 19, and was a dedicated and prolific mathematician for the rest of his life, working mainly in Turin and Berlin.

Joseph-Louis Lagrange (1736–1813)

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A portrait of Joseph-Louis Lagrange, who lived from 1736 to 1813. It shows a man with balding grey hair but dark eyebrows.

Joseph-Louis Lagrange (1736–1813)

To see how derivatives are written in Leibniz notation, consider the equation , which expresses a relationship between the variables and . You’ve seen that the formula for the gradient of the graph of the equation is

In Leibniz notation this equation is written as

So the notation means the same as , where . It’s read as ‘d y by d x’. When Leibniz notation is being used, is often referred to as the derivative of with respect to .

If you want to write the notation in a line of text, then you can write it as , just as you would do for a fraction. However, although the notation looks like a fraction, it’s important to remember that it isn’t a fraction!

Also, you should be aware that the ‘d’ that’s part of Leibniz notation has no meaning outside of it. In particular, although and look like and , respectively, the ‘d’ is certainly not a factor and must not be cancelled! In many mathematical texts, including this one, the ‘d’ in Leibniz notation appears in upright type, rather than the italic type used for variables, to emphasise this fact. (You don’t need to do anything special when you handwrite Leibniz notation – you should just write the ‘d’ in the normal way.)

To understand the thinking behind Leibniz notation, consider Figure 19. It’s exactly the same as Figure 17, which illustrates differentiation from first principles, except that some things are labelled differently. For example, the point at which we want to find the gradient is labelled instead of . This is because the emphasis here is on the relationship between the variables and , rather than on the idea of as a function of . Another difference is that the change in the -coordinate from the point at which we want to find the gradient to the second point is denoted by instead of , so the -coordinate of the second point is written as instead of .

The symbol is the lower-case Greek letter delta, and is read as ‘delta’. By convention, when the symbol is used as a prefix it indicates ‘a small change in’, so denotes a small change in . The change in the -coordinate from the point at which we want to find the gradient to the second point is denoted in a similar way, as , which means that the -coordinate of the second point is .

Figure 19 The points and on the graph of , and the line through them

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The points with coordinates (x, y) and (x + delta x, y + delta y) on the graph of y = f of x and the line through them. The x and y axes are shown but there are no scales. A smooth curve starts where both the coordinates are negative, rises to where the y coordinate is positive but the x coordinate is still negative and then turns smoothly to cut the positive y-axis and has positive coordinates for the remainder of the curve. It drops down a little but then turns smoothly, above the x-axis, and rises. Two points are marked on this last part of the curve. The first is labelled (x, y). A tangent to the curve is drawn in red at this point. The second point is above and to the right of the first. It is labelled (x + delta x, y + delta y). A blue line passes through the points (x, y) and (x + delta x, y + delta y). A blue horizontal dotted line, labelled delta x, starts at the point (x, y) and moves to the right, ending vertically below the point (x + delta x, y + delta y). From the end of this line a blue dotted vertical line rises to end at the point (x + delta x, y + delta y). It is labelled delta y.

Figure 19 The points and on the graph of , and the line through them

The gradient of the line that passes through the two points is then

As the second point gets closer and closer to the first point, the value of gets closer and closer to the gradient of the curve at the point . In other words, the gradient is the limit of as the second point gets closer and closer to the first point (and so as and get smaller and smaller). This is why the notation , which looks similar to , was chosen to represent the gradient.

The expression can be used instead of expression (2) to carry out differentiation from first principles, and you might see this done in some other texts on calculus. The process is exactly the same, just with replaced by , but it can look a bit more complicated at first sight.

Leibniz notation can be used in a variety of ways. For example, the symbol

means ‘the derivative with respect to of’. So, for example, a concise way to express the fact that the gradient of the graph of the equation is given by the formula is to write

As with Lagrange notation, Leibniz notation can be used with variable names other than the standard ones, and . For example,

and

Similarly (by Activity 5),

Sometimes, particularly on a computer algebra system, you might see written as

Usually, if a function is specified using function notation, then you use Lagrange notation for its derivative, whereas if it’s specified using an equation that expresses one variable in terms of another, then you use Leibniz notation. For example, if you know that , then you write , whereas if you know that , then you write . However, there are no absolute rules about this, and in fact it’s often helpful to mix the two notations. In particular, it’s often convenient to use the symbol , even when you’re mostly using Lagrange notation, as you’ll see. Lagrange notation and Leibniz notation are the two most common notations for derivatives, but there are other useful notations, including one invented by Isaac Newton. You’ll meet some of these notations if you go on to study calculus beyond this course, particularly in the area of applied mathematics.