Leibniz and the Rise of Differentials

In the autumn of 1675 a twenty-eight-year-old German polymath was puzzling over an old Greek problem: given an arbitrary curve and an arbitrary point on it, how do you draw the tangent line at that point? Euclid and Archimedes had solved it for circles and for a few conic sections, but nobody had a general method. Over the next decade Gottfried Wilhelm Leibniz built one, and in the process invented the notation $\frac{dy}{dx}$ that every calculus student still writes today.

This page follows that discovery in the order it actually happened. It begins with the geometric puzzle, moves through Leibniz’s experiments with the parabola and the secant lines that approximate a tangent, and reaches the moment when he realises that the ratio of two “infinitely small” quantities is the tangent slope. From there it follows the consequences: the notation, the rules of differentiation, the link to integration, and the applications to optics, mechanics and economics. It closes with the bitter priority dispute with Isaac Newton — the historical reason why so much of Leibniz’s work sat unread for a century, and the reason we still write $\frac{dy}{dx}$ rather than $\dot{y}$.

The Problem That Started It

The question that occupied Leibniz was this:

How can one draw the tangent to any curve at any point?

For a straight line the question is trivial — the tangent is the line itself. For a circle, Archimedes had a construction. But for a parabola, an ellipse, a cycloid, or one of the curves that the new geometry of the seventeenth century was producing, the direction of the tangent changes from point to point, and no general method existed for finding it.

Leibniz took the standard route of the scientific age: start from the simplest example, understand it completely, and then ask what the understanding teaches about the general case.

A Timeline of the Discovery

Experiment One: Tangents to the Parabola

Leibniz began with $y = x^2$, the simplest curve whose tangent genuinely changes direction.

Two facts are visible at once. The direction is not constant, so a “slope” of the curve has to be something that depends on the point. And the pattern is symmetric in an obvious way: the tangent at $(-1, 1)$ is the mirror image of the tangent at $(1, 1)$. What Leibniz wanted was a rule for the direction at an arbitrary point, not a table of guesses.

Experiment Two: Approximating the Tangent with Secants

The device that made the whole thing work is geometrically simple. Take two points on the curve, join them with a straight line — a secant — and let the second point slide towards the first. In the limit the secant becomes the tangent.

Leibniz fixed the point $(1, 1)$ and moved the second point towards it, recording the slope of each secant.

The slopes are heading somewhere definite. $3$, then $2.5$, then $2.1$, then $2.01$: the numbers are closing on $2$. The tangent at $(1,1)$ has slope $2$.

Leibniz was careful about the direction of the reasoning, and this is the conceptual heart of the story: the limit does not merely approximate the tangent slope, it is the tangent slope. The secant slope was always an approximation; the limit is the exact quantity the geometry was asking for.

The Breakthrough: Birth of the Differential Method

Now the same idea in symbols, for a general point $(x, x^2)$ of the parabola. Take a small increment $\Delta x$ and form the secant slope between $(x, x^2)$ and $(x + \Delta x, (x + \Delta x)^2)$:

$$\text{secant slope} = \frac{(x + \Delta x)^2 - x^2}{(x + \Delta x) - x}$$

Expand the numerator:

$$\text{secant slope} = \frac{x^2 + 2x\Delta x + \Delta x^2 - x^2}{\Delta x} = \frac{2x\Delta x + \Delta x^2}{\Delta x} = 2x + \Delta x$$

The algebra has done the crucial work: the secant slope is not some complicated expression but the simple linear quantity $2x + \Delta x$. Letting $\Delta x$ shrink to zero removes the leftover term:

$$\text{tangent slope} = \lim_{\Delta x \to 0} (2x + \Delta x) = 2x$$

Inventing the Notation

The calculation above works, but it leaves a lot of prose around the mathematics. Leibniz’s lasting contribution was to compress the whole procedure into a symbol, and then to make that symbol mean something.

He wrote $dx$ for an infinitely small increment of $x$ and $dy$ for the corresponding infinitely small increment of $y$, and he read $\frac{dy}{dx}$ as the ratio of the two. The letters have clear origins: $d$ is the first letter of the Latin differentia, and the integral sign $\int$ is a stretched, elongated version of the Latin summa for the capital Greek letter $\Sigma$.

Leibniz’s reading of this went further than the modern one. He did not regard $dx$ and $dy$ as convenient shorthand for limits; he regarded them as genuine infinitely small quantities, and $\frac{dy}{dx}$ as their exact ratio. That is why the notation is written as a fraction: Leibniz wanted the reader to see a genuine ratio, not a formal device. Modern calculus keeps the fraction notation but reinterprets it, a shift that took the better part of a century and two wars to settle.

The Geometric Meaning Deepens

The picture that emerged from the experiments can now be stated cleanly.

• The secant slope is the ratio of finite increments, $\frac{\Delta y}{\Delta x}$.
• The tangent slope is the ratio of infinitesimal increments, $\frac{dy}{dx}$.
• As $\Delta x \to 0$, the secant becomes the tangent.

Two consequences follow immediately. First, the tangent slope is the instantaneous rate of change of the function — the same object viewed as geometry and as motion. Second, the differential method is a generalisation of the geometry Leibniz had been reading: the tangent to a curve at a point is the limit of the secants, exactly as the area of a circle is the limit of inscribed polygons.

Checking the Result

Before building anything on top of it, Leibniz checked the rule against the numbers he had already computed by hand.

• At $x = 1$: the formula gives $2 \times 1 = 2$, which matches the secant slopes $3$, $2.5$, $2.1$, $2.01$ converging on $2$.
• At $x = 2$: the formula gives $2 \times 2 = 4$, and the tangent at $(2, 4)$ is indeed twice as steep as at $(1, 1)$ — consistent with the direction table in Experiment One.

From One Curve to All of Them

The moment the method is written in terms of $f$ rather than $x^2$, it applies to everything.

Tangent lines for any function. Given $y = f(x)$, the tangent slope at any point where the limit exists is $\lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$.

Extrema. A function’s extreme values occur where the tangent is horizontal, that is where $\frac{dy}{dx} = 0$. Refining this with a second derivative separates the two cases: $\frac{dy}{dx} = 0$ together with $\frac{d^2 y}{dx^2} < 0$ signals a local maximum, and $\frac{d^2 y}{dx^2} > 0$ signals a local minimum.

The shape of a curve. The sign of $\frac{dy}{dx}$ tells you whether the curve is rising or falling; the sign of $\frac{d^2 y}{dx^2}$ tells you whether it is convex upwards or downwards, and where its inflections lie.

Building the Rules of Differentiation

Working out the limit afresh for each new function would be hopeless. Leibniz therefore derived a small set of rules that generate all the others. The derivatives of the basic functions are

$$\frac{d}{dx} x^n = n x^{n-1}, \qquad \frac{d}{dx} e^x = e^x, \qquad \frac{d}{dx} \ln x = \frac{1}{x}, \qquad \frac{d}{dx} \sin x = \cos x, \qquad \frac{d}{dx} \cos x = -\sin x$$

and the combination rules are the sum and difference rule, the product rule, the quotient rule and the chain rule:

$$\frac{d(u+v)}{dx} = \frac{du}{dx} + \frac{dv}{dx}, \qquad \frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}, \qquad \frac{d\left(\frac{u}{v}\right)}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}, \qquad \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$

The Link to Integration

The 1686 paper contains Leibniz’s most important structural insight: differentiation and integration are inverse operations, and the notation expresses that structure literally.

• If $y = \int f(x)\,dx$, then $\frac{dy}{dx} = f(x)$.
• If $\frac{dy}{dx} = f(x)$, then $y = \int f(x)\,dx + C$.

Geometrically, differentiation measures the slope of a curve while integration measures the area under it, and the two are exact inverses: the slope of the area-curve recovers the original function, and the area under the derivative recovers the original curve up to a constant.

Where the New Tool Was Used

Optics. The path of a light ray is described by a function $y = f(x)$, and the law of refraction is a statement about the slope $\frac{dy}{dx}$ of that path at the point where the ray meets a boundary. Once you can differentiate a path function, you can state the refraction law.

Mechanics. Motion is described by a position function $s(t)$. Its derivative $\frac{ds}{dt}$ is the velocity, and its second derivative $\frac{d^2 s}{dt^2}$ is the acceleration. The whole of Newtonian mechanics is written in Leibniz’s notation.

Economics. A cost function $C(x)$ has marginal cost $\frac{dC}{dx}$, and a revenue function $R(x)$ has marginal revenue $\frac{dR}{dx}$. Leibniz applied the differential calculus to economic questions himself, and the $\int$ sign he invented was in use in the German states long before it was adopted in England.

Why the Notation Won

Three properties made Leibniz’s notation difficult to displace.

• Intuition. The fraction $\frac{dy}{dx}$ says at a glance that the derivative is a ratio, and it carries the geometric picture of a tangent slope with it. The English rival notation of Newton’s dots, $\dot{y}$ and $\ddot{y}$, says nothing about a ratio at all.
• Computability. Leibniz’s notation scales: it handles products, quotients, powers and compositions by rules that are stated once and then applied. A single rule like $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$ makes the composition of any two functions computable.
• Legacy. The notation spread with Leibniz’s international reputation, was carried into physics by the Continental school, and eventually became the universal language of the subject. Euler, Lagrange, Laplace, Gauss and Cauchy all wrote in it, and so does everyone since.

Historical Significance

Theoretical. Leibniz built a complete system of differential calculus on a clear geometric foundation, invented a notation that outlived every rival, and established the inverse relation between differentiation and integration that organises the whole subject.

Applied. His methods made it possible to attack problems in optics, mechanics, geometry and economics that had previously been out of reach, and the calculus he published in 1684 and 1686 was the first working general-purpose method for finding tangents, extrema and areas.

Pedagogical. The history of the notation is also a lesson in what a good symbol can do. Leibniz’s contemporaries were not convinced that infinitesimals were legitimate — George Berkeley’s The Analyst (1734) mocked them as the “ghosts of departed quantities” — and it took the reformulation of the derivative as a limit, in the nineteenth century, to make the calculus rigorous. The notation survived the change of foundations intact, which is the strongest compliment one can pay it.

The Priority Dispute

In October 1676, in London, Newton showed Leibniz his notes on fluxions; Leibniz showed Newton his method of differences. Each left with the conviction that the other had seen the essentials of an idea they had reached independently — Newton had begun his work in the 1660s, Leibniz his in 1675.

What followed was a slow escalation. Newton published nothing; Leibniz published in 1684 and 1686. In 1712 Newton wrote to the Royal Society, and the Society’s Commerium Epistolicum — a report of the evidence, of which Newton himself wrote a substantial part — concluded that the method of fluxions was Newton’s invention. Leibniz’s reply of 1713, the “Charta Volans”, pointed out that the Commerium was not a neutral report, and the argument degenerated into personal attack.

Leibniz died in 1716 with the dispute unresolved. The historical consensus reached in the nineteenth century is that the two men developed the calculus independently and essentially simultaneously, but that Leibniz had the better notation and was the only one of the two to publish promptly. The bitterness was not one-sided: Leibniz campaigned actively in German and French courts for recognition that Newton’s followers in England had failed to give him, and the English priority claim rested substantially on evidence assembled by Newton’s enemies as well as by Newton himself.

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Exercises

Exercise 1

Using Leibniz’s method, compute the derivative of $y = x^3$ at $x = 2$.

Exercise 2

Use Leibniz’s notation and rules to explain why $\frac{d}{dx}(x^2 + 3x) = 2x + 3$.

Exercise 3

Prove the product rule using Leibniz’s method, that is, starting from the limit definition of the derivative.

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Summary

Leibniz’s achievement was not a lucky formula but a method with a clear shape: approximate a tangent by a secant, simplify the secant slope algebraically, and pass to the limit. Because the reasoning used nothing specific to any curve, the same procedure worked for every function, and Leibniz could systematise it into a handful of rules. What made the method durable was the notation: $dx$, $dy$ and $\frac{dy}{dx}$ carried the geometric meaning of a ratio into every computation, and after the priority dispute of 1712–1716 the calculus split into a British school of dots and a Continental school of differentials — until the nineteenth century reformulated the derivative as a limit, at which point the difference dissolved and Leibniz’s notation won.

Symbols Used in This Article