Isaac Newton developed the mathematical methods behind calculus, which he called the method of fluxions, in the mid-1660s, while Cambridge University was closed because of plague, but he did not publish the work, it circulated only among a small circle of colleagues in manuscript form for decades. Gottfried Wilhelm Leibniz developed an independent version in the 1670s while working in Paris and Hanover, and published his results starting in 1684, using a system of notation, including the integral sign and the dy over dx notation for a derivative, that proved clearer and more flexible than Newton's, and remains the notation used almost everywhere calculus is taught today.
When Newton's supporters began accusing Leibniz of plagiarizing Newton's unpublished work in the early 1700s, what might have been a private academic disagreement became a public, nationalistic scandal. The Royal Society, of which Newton was president, appointed a committee in 1712 to investigate and rule on priority. The committee's report, Commercium Epistolicum, found in Newton's favor, a verdict that carried the Royal Society's institutional authority but was compromised from the start, since Newton himself wrote large portions of the supposedly neutral report anonymously and orchestrated much of its content.
The dispute poisoned relations between British and continental mathematicians for most of the eighteenth century, British mathematicians, out of loyalty to Newton, continued using his more awkward fluxion notation long after continental mathematicians, using Leibniz's system, had made faster progress, a self-inflicted setback historians of mathematics have long noted. The modern consensus, reached only once nationalist stakes had faded, is that Newton and Leibniz arrived at calculus independently, by different conceptual routes, at nearly the same historical moment, and that both deserve credit for the discovery.