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When you write dy/dx or ∫ x² dx, you are writing in the notation Gottfried Leibniz invented in Paris in 1675. Newton found calculus independently and earlier, but it is Leibniz's symbols that the world adopted. This post tells his story from reliable sources, shows why his notation is so useful, and works through the chain rule and a physics problem in his style.
Who was Gottfried Leibniz?
The MacTutor biography records that Leibniz was born in Leipzig on 1 July 1646 and died in Hannover on 14 November 1716. He entered the University of Leipzig at 14, where philosophy was well taught and mathematics, MacTutor notes, very poorly. He worked for most of his life as a diplomat, lawyer, librarian and historian for German courts, and did his mathematics alongside.
In 1672 he went to Paris on a diplomatic mission. In London in January 1673 he showed the Royal Society an unfinished calculating machine, and he was elected a Fellow that April. Christiaan Huygens then gave him a reading list in mathematics, and within a few years he went from a modest mathematician to the inventor of a calculus.
How did Leibniz invent the integral sign?
Leibniz first wrote "omn." (for omnia, "all") in front of the quantities he was adding up. According to Jeff Miller's Earliest Uses of Symbols of Calculus, hosted by MacTutor, he first used ∫ in an unpublished manuscript of 29 October 1675, writing that it would be useful to write ∫ for omn., "the sum of all the l's". The sign is a long s, for summa. Two weeks later, in a manuscript of 11 November 1675, he introduced dx, dy and dx/dy.
MacTutor adds that a manuscript of 21 November 1675 gives the product rule, and that by autumn 1676 he had d(xⁿ) = nxⁿ⁻¹ dx for whole and fractional n.
What did he publish, and why was there a dispute?
In 1684 Leibniz published Nova Methodus pro Maximis et Minimis in the Leipzig journal Acta Eruditorum, with the d notation and rules for powers, products and quotients. MacTutor notes it had no proofs, and Jacob Bernoulli called it an enigma rather than an explanation. The integral sign first appeared in print in his 1686 paper.
Newton's method of fluxions dated from 1671 but was not printed until 1736, and that delay led to a bitter priority dispute. In 1713 the Royal Society published a report, Commercium epistolicum, finding for Newton. MacTutor records that the committee never asked Leibniz for his side and that Newton himself wrote the report. Our post on Newton's three laws in the original covers Newton's side of the period.
Worked example: the chain rule in Leibniz's notation
Find dy/dx for y = (3x² + 1)⁵, and its value at x = 1.
Name the inside. Let u = 3x² + 1, so y = u⁵.
Differentiate each link. dy/du = 5u⁴ and du/dx = 6x.
Chain them. dy/dx = (dy/du)(du/dx) = 5u⁴ × 6x = 30x(3x² + 1)⁴.
Evaluate. At x = 1, u = 4, so dy/dx = 30 × 1 × 4⁴ = 30 × 256 = 7680.
The du "cancels" on paper, which is why the rule is easy to remember. It is not literally a fraction, but the notation reminds you which rates to multiply.
A physics use. A particle's speed depends on position as v = 2x (v in m/s, x in metres). Find its acceleration at x = 3 m. Acceleration is dv/dt = (dv/dx)(dx/dt) = v (dv/dx). Here dv/dx = 2, so a = 2x × 2 = 4x, giving 12 m/s² at x = 3 m. Our guide to the calculus you need for JEE physics has more problems of this type.
Common mistakes:
Forgetting the inner derivative, writing 5(3x² + 1)⁴ and stopping.
Treating dy/dx as a quotient you can split anywhere, for example "cancelling" d's in d²y/dx².
Using a = dv/dx instead of a = v dv/dx when v is given as a function of position.
What else did Leibniz work on?
MacTutor notes that he perfected a binary system of arithmetic by 1679 and published it in 1701, the base-2 system that digital computers use; our post on binary number systems covers the basics. He also worked on determinants, dynamics and philosophy.
What can a student take from Leibniz's way of working?
Good notation is half the thinking. He tried many symbols before settling on ∫ and d.
Start late, learn fast. He was a serious beginner in mathematics in his mid-twenties.
Publish and explain. His brief 1684 paper spread the method but puzzled readers; proofs and examples matter.
When the chain rule or a rate problem keeps going wrong in the same place, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can see which link of the chain you dropped.
Use Leibniz's chain rule to differentiate y = √(1 + x³), then find its value at x = 2.
