Infinitesimal Calculus: Meaning, History and Examples

What Is Infinitesimal Calculus?

Infinitesimal calculus is the name that differential and integral calculus carried when Isaac Newton and Gottfried Wilhelm Leibniz built it in the late 17th century, because its methods ran on infinitely small quantities called infinitesimals. Today the subject is simply called calculus, and the infinitesimals have been replaced in most textbooks by limits. The notation, though, never changed.

An infinitesimal is a quantity that is not zero yet is smaller in size than every positive real number. Written as a condition, a positive infinitesimal $\varepsilon$ would satisfy

$$0 < \varepsilon < \frac{1}{n} \quad \text{for every positive integer } n.$$

Here $\varepsilon$ is the infinitesimal and $n$ runs through $1, 2, 3, \ldots$ without end. With infinitesimals in hand, Leibniz described the two central operations of calculus like this:

$$\frac{dy}{dx} = \frac{f(x + dx) - f(x)}{dx}, \qquad \int_{a}^{b} f(x),dx = \text{the sum of infinitely many strips } f(x),dx.$$

In these formulas $dx$ is an infinitesimal change in $x$, $dy$ is the matching infinitesimal change in $y = f(x)$, and $f(x),dx$ is the area of one infinitely thin strip of height $f(x)$. The first formula gives the derivative (a rate of change); the second gives the integral (an accumulated total). One condition was always left unspoken: at the end of a derivative calculation, any term still containing $dx$ is thrown away. That rule worked, and it was also the weak spot critics aimed at.

Why Is Calculus Called Infinitesimal Calculus?

Calculus is called infinitesimal calculus because its first versions computed with infinitesimals, and the longer name stuck for about two centuries. The word "calculus" itself comes from the Latin for "pebble", since Romans counted with small stones, so a "calculus" was originally any system of reckoning. Leibniz named his two branches calculus differentialis and calculus summatorius, and "infinitesimal calculus" became the umbrella term for both.

Newton used a different name for the same ideas. He called his method the science of fluxions, and British schools kept that vocabulary into the 19th century. The full story of who discovered what, and when, is told on the history of calculus page; this article stays on the infinitesimals themselves.

Is An Infinitesimal A Real Number?

No real number is an infinitesimal, apart from zero, which the definition excludes. The reason is a property of the real numbers called the Archimedean property: for any positive real number $r$, there is a whole number $n$ large enough that $\frac{1}{n} < r$. So any positive real number fails the condition $0 < \varepsilon < \frac{1}{n}$ for some $n$.

That fact explains the whole drama of this subject in one line. Newton, Leibniz and Euler calculated with objects that the number line, as later mathematicians defined it, has no room for. Their answers were right, but the objects behind them had no rigorous definition until the 20th century.

A Physics Forums thread, "The infinitesimal in Calculus", tries to define an infinitesimal $d$ as a positive number smaller than every other positive number and asks whether that definition is correct. Inside the real numbers it cannot be: if $d > 0$ were the smallest positive real, then $\frac{d}{2}$ would be a smaller positive real, which is a contradiction.

Three properties early users relied on, all stated informally:

• An infinitesimal is not zero, so you may divide by it.

• An infinitesimal is smaller than any ordinary quantity, so $x + dx$ is "the same point" as $x$ for practical purposes.

• A product of two infinitesimals is negligible compared with one infinitesimal, so terms like $(dx)^2$ can be dropped beside $dx$.

The first and third properties pull against each other. Division needs $dx \neq 0$, while dropping terms treats $dx$ almost as if it were $0$.

How Did Leibniz Calculate With Dx And Dy?

Leibniz calculated by writing every change as a differential, expanding, and discarding higher-order infinitesimals. According to the MacTutor biography, he first wrote the integral sign together with $dx$ in a manuscript dated 21 November 1675, and he published his differential calculus in 1684 in the journal Acta Eruditorum, under the title Nova Methodus pro Maximis et Minimis. His integral sign appeared in print in 1686.

His derivation of the product rule shows the whole method. Let $x$ and $y$ change by $dx$ and $dy$. The product $xy$ changes by

$$d(xy) = (x + dx)(y + dy) - xy$$

$$d(xy) = x,dy + y,dx + dx,dy.$$

The last term $dx,dy$ is a product of two infinitesimals, so Leibniz discarded it:

$$d(xy) = x,dy + y,dx.$$

Divide by $dx$ and you have the modern product rule. The method is fast, it is easy to remember, and it gives correct results for every function Leibniz worked with.

Newton worked differently. He pictured quantities flowing in time and wrote their rates (fluxions) with a dot, as in $\dot{x}$, using a small increment he called "o" during the calculation and then letting it vanish. The two methods produce identical results, and Leibniz's $dx$ notation is the one that survived worldwide.

Why Did Limits Replace Infinitesimals?

Limits replaced infinitesimals because nobody in the 18th century could say exactly what an infinitesimal was, and the limit gave calculus a definition built only from ordinary real numbers. The pressure began in 1734, when George Berkeley published The Analyst and argued that calculus divided by a quantity and then treated it as zero, a shift in the hypothesis. He called the vanishing increments "ghosts of departed quantities".

Mathematicians answered in stages:

• Bernard Bolzano (1817) sketched the epsilon-delta idea for continuity, though his work stayed little known in his lifetime.

• Augustin-Louis Cauchy (1821) wrote the Cours d'analyse for students at the Ecole Polytechnique, developing the basic theorems of calculus as rigorously as he could, with limits and infinitesimals described through variable quantities.

• Karl Weierstrass (1861) stated the epsilon-delta definition of limit in the form still taught today, through his Berlin lectures.

The Weierstrass definition contains no infinitesimal at all. A function $f$ has the limit $L$ as $x$ approaches $a$ when

$$\text{for every } \varepsilon > 0 \text{ there is a } \delta > 0 \text{ such that } 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon.$$

Here $\varepsilon$ and $\delta$ are ordinary positive real numbers, as small as you like but never infinitely small. The derivative then becomes a limit of a function, the difference quotient as $h$ shrinks:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$$

Compare this with Leibniz's version. The limit never sets $h$ equal to zero and never divides by zero, which is exactly the gap Berkeley spotted. The full development of this formula is on the definition of the derivative page.

How Did Abraham Robinson Bring Infinitesimals Back?

Abraham Robinson brought infinitesimals back in 1961 by building a number system, the hyperreal numbers, that contains the real numbers plus genuine infinitesimals and infinite numbers, all obeying the same algebra. He set out the theory in his 1966 book Non-standard Analysis. For the first time, Leibniz's style of reasoning had a foundation as rigorous as Weierstrass's.

The key tool is the standard part. Every finite hyperreal number $r$ sits infinitely close to exactly one real number, written $\operatorname{st}(r)$. The derivative then reads almost like Leibniz:

$$f'(x) = \operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right) \quad \text{for any nonzero infinitesimal } dx.$$

Taking the standard part is the rigorous version of "throw away the leftover $dx$". It replaces the old rule of thumb with a defined operation, so Berkeley's objection no longer applies.

A related Physics Forums thread asks how hyperreal numbers make infinitesimals logically sound in calculus. Robinson's answer is the transfer principle: a statement about the reals written in first-order logic is true exactly when its hyperreal version is true, so the familiar algebra carries over unchanged. The Archimedean property still transfers, but with hyperintegers in place of ordinary whole numbers, which is how infinitesimals fit in without breaking any rule.

The distinction between "an infinitesimal does not exist among the reals" and "an infinitesimal exists in a larger system" takes a while to settle; it is easy to read Robinson's work as proving the reals were wrong, when it only shows they can be extended. In 1976 H. Jerome Keisler published Elementary Calculus: An Infinitesimal Approach, a first-year textbook built on hyperreals, and he now keeps it freely available online. Other systems with infinitesimals exist too, such as dual numbers (used in automatic differentiation) and smooth infinitesimal analysis, but most courses still teach the limit approach.

When Can You Treat A Derivative Like A Fraction?

You can treat $\frac{dy}{dx}$ like a fraction whenever a single independent variable is involved, because those manipulations are backed by theorems. This is one of the most-viewed calculus questions on Math Stack Exchange ("Is dy/dx not a ratio?"), and the honest answer is: in the limit definition it is not a ratio, in Leibniz's and Robinson's frameworks it is, and in everyday one-variable work the fraction-style moves give correct results.

Two places where the fraction picture works:

• The chain rule. Writing $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ looks like cancelling $du$, and the result is a true theorem.

• Inverse functions. $\frac{dx}{dy} = \frac{1}{dy/dx}$ wherever $\frac{dy}{dx} \neq 0$, again matching the fraction picture.

A second Stack Exchange question asks when not to treat $\frac{dy}{dx}$ as a fraction. Trouble starts with several variables. For partial derivatives of $x + y + z = 0$, "cancelling" suggests

$$\frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} = 1,$$

but each factor equals $-1$, so the product is actually $-1$. Partial derivatives each hold a different variable fixed, so their symbols are not fractions with a shared meaning.

How Is Dx Used In Calculus Today?

Today $dx$ has three jobs: it names the variable of integration, it serves as a differential (a real number standing for a small change) in approximations, and it lets you separate variables in differential equations. Each job descends directly from infinitesimal calculus.

• In an integral. Math Stack Exchange's "What is dx in integration?" gets the infinitesimal answer historically (the width of a strip) and the modern answer formally (a marker of which variable the Riemann sum runs over). The area under a curve is the limit of sums of $f(x_k),\Delta x$ strips, and $dx$ is the descendant of $\Delta x$.

• As a differential. With $y = f(x)$, the differential $dy = f'(x),dx$ gives a linear estimate of the change in $y$ for a small real change $dx$.

• In substitution. In integration by substitution, writing $u = x^2$ and $du = 2x,dx$ treats $du$ and $dx$ as differentials, and the bookkeeping comes out right.

• In physics. Physicists "multiply both sides by $dx$" when separating variables, a habit another Stack Exchange thread asks about. The move is justified by the chain rule and the substitution rule, which is why the answers come out correct.

Examples Of Infinitesimal Calculus

Each example works the same problem twice: first with infinitesimals in Leibniz's style (made rigorous by the standard part), then with limits. The two columns always agree.

Example 1

Find the derivative of $f(x) = x^2$.

Infinitesimal version:

$$\frac{(x + dx)^2 - x^2}{dx} = \frac{2x,dx + (dx)^2}{dx}$$

$$= 2x + dx$$

$$\operatorname{st}(2x + dx) = 2x$$

Limit version:

$$\lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \to 0} (2x + h)$$

$$= 2x$$

Final answer: $f'(x) = 2x$. Geometrically, the tangent to the parabola at $x = 1$ has slope $2$.

Example 2

Find the slope of $f(x) = x^3$ at $x = 2$.

A tempting shortcut is to treat $dx$ as a small ordinary number, say $0.1$, and compute directly:

$$\frac{(2.1)^3 - 8}{0.1} = \frac{9.261 - 8}{0.1}$$

$$= 12.61$$

That cannot be the slope. Try $0.001$ instead and the quotient becomes $12.006001$, a different number. A slope at a point has one value, so a method whose answer depends on the size of the step is measuring an average over a small interval, not the slope at $x = 2$. The fix is to keep $dx$ symbolic and remove what is left only at the end:

$$\frac{(2 + dx)^3 - 8}{dx} = \frac{12,dx + 6,(dx)^2 + (dx)^3}{dx}$$

$$= 12 + 6,dx + (dx)^2$$

$$\operatorname{st}\big(12 + 6,dx + (dx)^2\big) = 12$$

The limit version agrees: $\lim_{h \to 0} (12 + 6h + h^2) = 12$. The numbers $12.61$ and $12.006001$ were the leftover terms $6,dx + (dx)^2$ evaluated at real step sizes.

Final answer: the slope is $12$.

Example 3

Find the derivative of $\sin x$.

Infinitesimal version. Use the angle-addition formula, then the first-order behaviour of sine and cosine for an infinitesimal angle ($\sin dx \approx dx$ and $\cos dx \approx 1$, each with an error of second order):

$$\sin(x + dx) - \sin x = \sin x \cos(dx) + \cos x \sin(dx) - \sin x$$

$$\approx \sin x \cdot 1 + \cos x \cdot dx - \sin x$$

$$= \cos x,dx$$

Dividing by $dx$ and taking the standard part gives $\cos x$.

Limit version. The same expansion needs two standard limits:

$$\lim_{h \to 0} \frac{\sin h}{h} = 1, \qquad \lim_{h \to 0} \frac{\cos h - 1}{h} = 0$$

$$\lim_{h \to 0} \frac{\sin(x + h) - \sin x}{h} = \sin x \cdot 0 + \cos x \cdot 1$$

$$= \cos x$$

Final answer: $\frac{d}{dx}(\sin x) = \cos x$.

Example 4

Find the area under $y = x^2$ from $x = 0$ to $x = 1$ as a sum of thin strips.

Split $[0, 1]$ into $n$ strips of width $\frac{1}{n}$ and use the right-hand height $\left(\frac{k}{n}\right)^2$ for strip $k$:

$$S_n = \sum_{k=1}^{n} \left(\frac{k}{n}\right)^2 \cdot \frac{1}{n}$$

$$= \frac{(n + 1)(2n + 1)}{6n^2}$$

Limit version: let $n \to \infty$.

$$\lim_{n \to \infty} \frac{(n + 1)(2n + 1)}{6n^2} = \frac{2}{6}$$

$$= \frac{1}{3}$$

Infinitesimal version: take $n$ infinitely large, so each strip has infinitesimal width $dx = \frac{1}{n}$, and the standard part of $S_n$ is $\frac{1}{3}$. With $n = 10$ the sum is $0.385$, and with $n = 100$ it is $0.33835$, closing in on $\frac{1}{3}$.

Final answer: $\int_{0}^{1} x^2,dx = \frac{1}{3}$.

Example 5

Solve $\frac{dy}{dx} = ky$, where $k$ is a constant, by separating the differentials.

Treat $dy$ and $dx$ as separate quantities, the way Leibniz would:

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

$$\int \frac{1}{y},dy = \int k,dx$$

$$\ln|y| = kx + C$$

$$y = Ae^{kx}$$

Here $A = \pm e^{C}$ (and $A = 0$ covers the solution $y = 0$). The limit-based justification is the chain rule applied to $\ln|y(x)|$, whose derivative is $\frac{y'}{y} = k$. This is the standard method for a separable differential equation.

Final answer: $y = Ae^{kx}$.

Example 6

Estimate $\sqrt{4.1}$ with the differential $dy = f'(x),dx$.

Take $f(x) = \sqrt{x}$, $x = 4$ and $dx = 0.1$. Then $f'(x) = \frac{1}{2\sqrt{x}}$, so

$$dy = \frac{1}{2\sqrt{4}} \cdot 0.1$$

$$= 0.025$$

$$\sqrt{4.1} \approx 2 + 0.025 = 2.025$$

The true value is $2.0248$ to 4 decimal places. Here $dx$ is an ordinary small number, not an infinitesimal, and that is why the answer is an estimate rather than exact. Geometrically, $dy$ is the rise along the tangent line, while the true change is the rise along the curve.

Final answer: $\sqrt{4.1} \approx 2.025$.

Why Did Mathematicians Need Infinitesimals In The First Place?

"Two problems that ordinary algebra could not crack" is the short reason infinitesimals were invented. Seventeenth-century scientists needed:

• The speed of a moving body at one instant. Average speed divides distance by time, but at a single instant both are zero, and $\frac{0}{0}$ says nothing. An infinitesimal time step gave a nonzero denominator.

• The area of a curved region. Rectangles fit straight edges only. Slicing a curved region into infinitely thin strips turned area into a sum, the idea Archimedes approached with his method of exhaustion and that Johannes Kepler used on wine barrels.

Infinitesimals turned both questions into calculations. The two answers then turned out to be inverse operations, the content of the fundamental theorem of calculus, which is why one subject and one notation cover rates and totals at once. Planetary orbits, falling bodies and later electrical circuits were all described with the same $dx$.

Where Do Students Trip Up On Infinitesimal Calculus?

The mistakes below come from mixing up the three meanings of $dx$: the historical infinitesimal, the modern limit, and the everyday differential.

Mistake 1: Treating The Step As A Tiny Ordinary Number

Where it slips in: Finding a derivative by plugging in a very small step such as $0.001$ and calling the result exact.

Don't do this: Report $12.006001$ as the slope of $x^3$ at $x = 2$.

The correct way: Keep the step symbolic, simplify, and only then let it vanish (limit) or take the standard part (hyperreals); the slope is exactly $12$.

The rusher tends to make this one, because a calculator answer feels finished. Computers face the same issue: with $f(x) = x^2$ at $x = 1$, the quotient $\frac{(1 + h)^2 - 1}{h}$ in standard floating-point arithmetic gives about $2.0001$ for $h = 0.0001$ but exactly $0$ for $h = 10^{-17}$, because the machine rounds $1 + h$ back to $1$.

Mistake 2: Cancelling Differentials Everywhere

Where it slips in: Multivariable problems, after a student has learned the chain rule as "the $du$'s cancel".

Don't do this: Assume $\frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} = 1$.

The correct way: Remember each partial derivative holds a different variable fixed; for $x + y + z = 0$ the product is $-1$.

The memorizer who learned "fractions cancel" as a rule, rather than the chain rule as a theorem, freezes here. A student who can say why the chain rule holds has no trouble seeing where it stops.

Mistake 3: Dropping The Differential From An Integral

Where it slips in: Substitution problems, where the $dx$ is treated as decoration.

Don't do this: Write $\int \cos(2x),dx = \sin(2x) + C$ after setting $u = 2x$ and ignoring how $dx$ changes.

The correct way: Convert the differential too: $du = 2,dx$, so $dx = \frac{du}{2}$ and $\int \cos(2x),dx = \frac{\sin(2x)}{2} + C$.

Students who write the $dx$ in every integral, every time, stop losing that factor of $\frac{1}{2}$; the differential carries the conversion for them. A quick check is to differentiate the answer and see the integrand come back.

The first of these mistakes has a famous real-world version. The step "divide by $dx$, then pretend it is $0$" is exactly what George Berkeley attacked in The Analyst in 1734, arguing that calculus reached true results through a logical fallacy. His criticism started the search for rigour that ended with Cauchy's and Weierstrass's limits.

Practice Problems On Infinitesimal Calculus

1. Use the infinitesimal method, expanding $5(x + dx)^2$, to find the derivative of $5x^2$.

2. For $y = x^3 - x$, find the differential $dy$, then evaluate it at $x = 2$ with $dx = 0.01$.

3. Use $\frac{1}{x + dx} - \frac{1}{x}$ to find the derivative of $\frac{1}{x}$.

4. Write the right-hand strip sum for $\int_{0}^{2} x,dx$ with $n$ strips, simplify it, and find its limit.

5. Separate the differentials to solve $\frac{dy}{dx} = 3y$ with $y(0) = 2$.

6. Evaluate $\int x e^{x^2},dx$ using $u = x^2$.

Answer to Question 1: $10x$.

Answer to Question 2: $dy = (3x^2 - 1),dx$, which is $0.11$ at $x = 2$, $dx = 0.01$.

Answer to Question 3: the difference is $\frac{-dx}{x(x + dx)}$, so the derivative is $-\frac{1}{x^2}$.

Answer to Question 4: the sum is $2 + \frac{2}{n}$, with limit $2$.

Answer to Question 5: $y = 2e^{3x}$.

Answer to Question 6: $\frac{e^{x^2}}{2} + C$.

Key Takeaways

• Infinitesimal calculus is the original name for calculus, built on quantities smaller than every positive real number.

• Leibniz's $dx$ and $dy$ notation, first written in 1675, survived even after the infinitesimals behind it were replaced.

• Berkeley's 1734 critique led to the limit definitions of Cauchy (1821) and Weierstrass (1861).

• Robinson's hyperreal numbers (1961) made infinitesimals rigorous, with the standard part replacing "throw away the leftover $dx$".

• Never replace $dx$ with a small ordinary number when you want an exact derivative.

Where To Go From Here

Work through practice problems 1 to 3 using both the infinitesimal method and the limit method; if the leftover $dx$ terms confuse you, return to Example 2 and the section on how Leibniz calculated with $dx$ and $dy$. For a live session that works through these two views of the derivative, book a free demo with a Bhanzu trainer:

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