Leibniz's notation

dy

dx

d ²y

dx²

Gottfried Wilhelm von Leibniz (1646 - 1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.^[1]

Consider y as a function of a variable x, or y = f(x). If this is the case, then the derivative of y with respect to x, which later came to be viewed as the limit

$\lim _{{\Delta x\rightarrow 0}}{\frac {\Delta y}{\Delta x}}=\lim _{{\Delta x\rightarrow 0}}{\frac {f(x+\Delta x)-f(x)}{\Delta x}},$

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

$\frac{dy}{dx}=f'(x),$

where the right hand side is Joseph-Louis Lagrange's notation for the derivative of f at x. 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:

$f'(x)={\rm st}\Bigg( \frac{\Delta y}{\Delta x} \Bigg)$.

Then one sets $dx=\Delta x$, $dy = f'(x) dx\,$, so that by definition, $f'(x)\,$ is the ratio of dy by dx.

Similarly, although mathematicians sometimes now view an integral

$\int f(x)\,dx$

as a limit

$\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,$

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

History

The Newton–Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the $\int$ character. He based the character on the Latin word summa ("sum"), which he wrote ſumma with the elongated s commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686,^[2] but he had been using it in private manuscripts at least since 1675.^[3] Leibniz first used $dx$ in an article also in Acta Eruditorum in 1684.^[4]

English mathematicians were encumbered by Newton's dot notation until 1803 when Robert Woodhouse published a description of the continental notation. Later the Analytical Society at Cambridge University promoted the adoption of Leibniz's notation.

At the end of the 19th century, Weierstrass's followers ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above, while Cauchy exploited both infinitesimals and limits (see Cours d'Analyse). Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f(x) as measured in meters per second, and dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.

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 non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based on Robinson's approach.

Leibniz's notation for differentiation

Suppose we have a variable y representing a function f of a variable x:

$y=f(x)\,$

Then we can write the derivative of the function f(x), in Leibniz's notation for differentiation, as the following:

$\frac{dy}{dx}\,\text{ or }\frac{d}{dx}y\,\text{ or }\frac{d\bigl(f(x)\bigr)}{dx}\,$

The Leibniz notation expression dy/dx is sometimes expressed in Lagrange's notation as the following:

${\frac {dy}{dx}}\,=y'$

Lagrange's "prime" notation can also be used for the equation where f(x) is substituted for y:

${\frac {d{\bigl (}f(x){\bigr )}}{dx}}=f'(x)\,$

The Lagrange notation f ′(x) (read as "f prime of x") is a common way to express the derivative function. Note that we can also use Newton's notation, which is often used for derivatives with respect to time (like velocity) and requires placing a dot over the dependent variable (in this case, x):

${\frac {dx}{dt}}={\dot {x}}\,$

The expression dy/dx should not be read as the division of two quantities dx and dy; rather, the whole expression can be seen as a single symbol that is shorthand for

$\lim_{\delta x \to 0} \frac{\delta y}{\delta x}$

(note δ vs. d). Alternatively, it can 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. However, the division-like notation is useful because in many situations, the derivative operator does behave like a division, making some results about derivatives easy to obtain and remember.^[5]

Leibniz notation for higher derivatives

For higher derivatives, we express them as follows:

$\frac{d^n\bigl(f(x)\bigr)}{dx^n}\text{ or }\frac{d^ny}{dx^n}$

denotes the nth derivative of f(x) or y respectively. For example, the first derivative can be written as dy/dx, the second derivative as d²y/dx², and so on.

This notation (with the second derivative as an example ) is derived from the following formula:

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

A more complicated example is the third derivative:

$\frac{d \left(\frac{d \left( \frac{d \left(f(x)\right)} {dx}\right)} {dx}\right)} {dx}\,,$

which we can loosely write as:

$\left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr) = \frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)\,.$

Now drop the parentheses and we have:

$\frac{d^3}{dx^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{d^3y}{dx^3}\,.$

Chain rule

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

$\frac{dy}{dx} = \frac{dy}{du_1} \cdot \frac{du_1}{du_2} \cdot \frac{du_2}{du_3}\cdots \frac{du_n}{dx}\,,$

etc., and:

$\int y \, dx = \int y \frac{dx}{du} \, du.$

See also

• Notation for differentiation
• Newton's notation
• Leibniz and Newton calculus controversy

Notes

1. Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
2. Stillwell, John (1989). Mathematics and its History. Springer. p. 110.
3. Child, J. M. (1920). Early Mathematical Manuscripts of Leibniz. Open Court. pp. 73–74, 80.
4. Cajori, Florian (1993) [1928]. A History of Mathematical Notations. New York: Dover. p. 204. ISBN 0-486-67766-4.
5. Jordan, D. W.; Smith, P. (2002). Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences. Oxford University Press. p. 58.