Derivative

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Ask what the speed of a falling object is at one exact instant, and ordinary language runs into a wall: speed is distance divided by time, and at a single instant no time has passed for anything to divide by. The derivative is the answer mathematics eventually gave to that problem, and it took most of the seventeenth century to arrive at it cleanly. The trick, in outline, is to ask the question over a very short interval instead of an instant, get an answer, then ask what happens to that answer as the interval shrinks toward zero without ever quite reaching it. Newton called the result a fluxion; Leibniz called it a differential and by 1675 had settled on the dy/dx notation still taught today. Both men treated the shrinking interval with a looseness that drew sharp criticism, most famously in Bishop Berkeley's 1734 attack, and efforts to tighten the reasoning continued for a century afterward; it was Cauchy, in the nineteenth century, who built the rigorous basis the calculus had been missing, and Weierstrass's own lectures a few decades later took up the same foundational project. What survived the two centuries of repair is exactly the intuition Newton and Leibniz started with: a derivative is simply the slope of a curve at one single point, made precise.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Isaac Newton worked out the essentials of the derivative in private notes dating to the mid-1660s and did not publish them for decades. Gottfried Leibniz, working independently in Germany, published his own version, with the dy/dx notation still in use today, in 1684, and for a while nobody on the continent had reason to think Newton had gotten there first. When the Royal Society, of which Newton was president, formally investigated the question of priority in 1712, it found in Newton's favor, which surprises nobody once you notice who was doing the investigating. Historians today generally accept that both men reached the calculus independently, working from different starting intuitions (Newton's from physics and motion, Leibniz's from a more purely symbolic, algebraic approach) and that the accusations of plagiarism thrown by both sides' supporters, which poisoned relations between British and continental mathematics for most of the eighteenth century, were unfounded. The practical cost was real: British mathematicians, loyal to Newton's notation out of national pride, fell behind their continental counterparts, who adopted Leibniz's more flexible dy/dx system and used it to develop calculus faster. The dispute is exactly the kind of priority argument the atlas records as a dissent rather than settling by fiat; crediting one side alone is a convention, not a finding.