Leibniz's notation

Leibniz's notation for differentiation

Suppose a dependent variable y represents a function f of an independent variable x, that is,

${\displaystyle y=f(x).}$

Then the derivative of the function f, in Leibniz's notation for differentiation, can be written as

${\displaystyle \frac{dy}{dx}\text{ or }\frac{d}{dx}y\text{ or }\frac{d(f(x))}{dx}.}$

The Leibniz expression, also, at times, written dy/dx, is one of several notations used for derivatives and derived functions. A common alternative is Lagrange's notation

${\displaystyle \frac{dy}{dx}=y^{\prime }=f^{\prime }(x).}$

Another alternative is Newton's notation, often used for derivatives with respect to time (like velocity), which requires placing a dot over the dependent variable (in this case, x):

${\displaystyle \frac{dx}{dt}=\dot{x}.}$

Lagrange's "prime" notation is especially useful in discussions of derived functions and has the advantage of having a natural way of denoting the value of the derived function at a specific value. However, the Leibniz notation has other virtues that have kept it popular through the years.

In its modern interpretation, the expression ⁠dy/dx⁠ should not be read as the division of two quantities dx and dy (as Leibniz had envisioned it); rather, the whole expression should be seen as a single symbol that is shorthand for

${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}$

(note Δ vs. d, where Δ indicates a finite difference).

The expression may also be thought of as the application of the differential operator ⁠d/dx⁠ (again, a single symbol) to y, regarded as a function of x. This operator is written D in Euler's notation. Leibniz did not use this form, but his use of the symbol d corresponds fairly closely to this modern concept.

While there is traditionally no division implied by the notation (but see Nonstandard analysis), the division-like notation is useful since in many situations, the derivative operator does behave like a division, making some results about derivatives easy to obtain and remember. This notation owes its longevity to the fact that it seems to reach to the very heart of the geometrical and mechanical applications of the calculus.

Leibniz notation for higher derivatives

If y = f(x), the nth derivative of f in Leibniz notation is given by,

${\displaystyle f^{(n)}(x)=\frac{d^{n}y}{d{x}^n}.}$

This notation, for the second derivative, is obtained by using ⁠d/dx⁠ as an operator in the following way,

${\displaystyle \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right).}$

A third derivative, which might be written as,

${\displaystyle \frac{d\left(\frac{d\left(\frac{dy}{dx}\right)}{dx}\right)}{dx},}$

can be obtained from

${\displaystyle \frac{d^{3}y}{d{x}^3}=\frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)=\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right).}$

Similarly, the higher derivatives may be obtained inductively.

While it is possible, with carefully chosen definitions, to interpret ⁠dy/dx⁠ as a quotient of differentials, this should not be done with the higher order forms. However, an alternative Leibniz notation for differentiation for higher orders allows for this.

This notation was, however, not used by Leibniz. In print he did not use multi-tiered notation nor numerical exponents (before 1695). To write x³ for instance, he would write xxx, as was common in his time. The square of a differential, as it might appear in an arc length formula for instance, was written as dxdx. However, Leibniz did use his d notation as we would today use operators, namely he would write a second derivative as ddy and a third derivative as dddy. In 1695 Leibniz started to write d²⋅x and d³⋅x for ddx and dddx respectively, but l'Hôpital, in his textbook on calculus written around the same time, used Leibniz's original forms.

Leibniz's notation for integration

Leibniz introduced the integral symbol for integration (or "antidifferentiation") now commonly used today:${\displaystyle {\displaystyle \int }}$

The notation was introduced in 1675 in his private writings; it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686. The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.

Use in various formulas

One reason that Leibniz's notations in calculus have endured so long is that they permit the easy recall of the appropriate formulas used for differentiation and integration. For instance, the chain rule—suppose that the function g is differentiable at x and y = f(u) is differentiable at u = g(x). Then the composite function y = f(g(x)) is differentiable at x and its derivative can be expressed in Leibniz notation as,

${\displaystyle \frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}.}$

This can be generalized to deal with the composites of several appropriately defined and related functions, u₁, u₂, ..., u_n and would be expressed as,

${\displaystyle \frac{dy}{dx}=\frac{dy}{d{u}_1}\cdot \frac{d{u}_1}{d{u}_2}\cdot \frac{d{u}_2}{d{u}_3}\cdots \frac{d{u}_n}{dx}.}$

Also, the integration by substitution formula may be expressed by

${\displaystyle \int ydx=\int y\frac{dx}{du}du,}$

where x is thought of as a function of a new variable u and the function y on the left is expressed in terms of x while on the right it is expressed in terms of u.

If y = f(x) where f is a differentiable function that is invertible, the derivative of the inverse function, if it exists, can be given by,

${\displaystyle \frac{dx}{dy}=\frac{1}{\left(\frac{dy}{dx}\right)},}$

where the parentheses are added to emphasize the fact that the derivative is not a fraction.

However, when solving differential equations, it is easy to think of the dys and dxs as separable. One of the simplest types of differential equations is

${\displaystyle M(x)+N(y)\frac{dy}{dx}=0,}$

where M and N are continuous functions. Solving (implicitly) such an equation can be done by examining the equation in its differential form,

${\displaystyle M(x)dx+N(y)dy=0}$

and integrating to obtain

${\displaystyle \int M(x)dx+\int N(y)dy=C.}$

Rewriting, when possible, a differential equation into this form and applying the above argument is known as the separation of variables technique for solving such equations.

In each of these instances the Leibniz notation for a derivative appears to act like a fraction, even though, in its modern interpretation, it isn't one.

Modern justification of infinitesimals

In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy Łoś, Abraham Robinson developed mathematical explanations for Leibniz's infinitesimals that were acceptable by contemporary standards of rigor, and developed nonstandard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year calculus textbook, Elementary calculus: an infinitesimal approach, based on Robinson's approach.

From the point of view of modern infinitesimal theory, Δx is an infinitesimal x-increment, Δy is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:

${\displaystyle f^{\prime }(x)=\mathrm{s}\mathrm{t}(\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x})}$.

Then one sets ${\displaystyle dx=\mathrm{\Delta }x}$, ${\displaystyle dy=f^{\prime }(x)dx}$, so that by definition, ${\displaystyle f^{\prime }(x)}$ is the ratio of dy by dx.

Similarly, although most mathematicians now view an integral

${\displaystyle \int f(x)dx}$

as a limit

${\displaystyle \underset{\mathrm{\Delta }x\to 0}{lim}\sum _{i}f(x_i)\mathrm{\Delta }x,}$

where Δx is an interval containing x_i, Leibniz viewed it as the sum (the integral sign denoted summation for him) of infinitely many infinitesimal quantities f(x) dx. From the viewpoint of nonstandard analysis, it is correct to view the integral as the standard part of such an infinite sum.

The trade-off needed to gain the precision of these concepts is that the set of real numbers must be extended to the set of hyperreal numbers.

Other notations of Leibniz

Leibniz experimented with many different notations in various areas of mathematics. He felt that good notation was fundamental in the pursuit of mathematics. In a letter to l'Hôpital in 1693 he says:

One of the secrets of analysis consists in the characteristic, that is, in the art of skilful employment of the available signs, and you will observe, Sir, by the small enclosure [on determinants] that Vieta and Descartes have not known all the mysteries.

He refined his criteria for good notation over time and came to realize the value of "adopting symbolisms which could be set up in a line like ordinary type, without the need of widening the spaces between lines to make room for symbols with sprawling parts." For instance, in his early works he heavily used a vinculum to indicate grouping of symbols, but later he introduced the idea of using pairs of parentheses for this purpose, thus appeasing the typesetters who no longer had to widen the spaces between lines on a page and making the pages look more attractive.

Many of the over 200 new symbols introduced by Leibniz are still in use today. Besides the differentials dx, dy and the integral sign ( ∫ ) already mentioned, he also introduced the colon (:) for division, the middle dot (⋅) for multiplication, the geometric signs for similar (~) and congruence (≅), the use of Recorde's equal sign (=) for proportions (replacing Oughtred's :: notation) and the double-suffix notation for determinants.