History and applications
The Newton–Leibniz controversy
Newton described his version of differential calculus as 'the method of fluxions'. He wrote a paper on fluxions in 1666, but like many of his works, it was not published until decades later. His magnum opus Philosophiae naturalis principia mathematica (Mathematical principles of natural philosophy) was published in 1687. This work includes his theories of motion and gravitation, but does not include much calculus explicitly — although there is some explanation of calculus at the beginning, and Newton certainly used calculus to formulate his theories. Nonetheless, Newton's 'method of fluxions' did not explicitly appear in print until 1693.
Leibniz, on the other hand, published his first paper on calculus in 1684 — and claimed to have discovered calculus in the 1670s. From the published record, at least, Leibniz seemed to have discovered calculus first.
While Newton and Leibniz initially had a cordial relationship, Leibniz and his followers did not take kindly to a statement made by the English mathematician John Wallis. With a rather xenophobic and quarrelsome character, Wallis fought priority disputes on behalf of English scientists throughout his life. In 1695, perhaps inadvertently, Wallis intimated that Leibniz learned about calculus from Newton — a claim now known to be false.
Then, offended by a statement of Leibniz that certain mathematical problems could only be solved by Leibniz's own version of the calculus, a mathematician named Fatio de Duiller in 1699 accused Leibniz of plagiarism. Things only went downhill from there. It did not help matters that Newton and Leibniz also disagreed on philosophical questions.
In 1712 the Royal Society in England wrote a report purporting to settle the matter — except, the whole investigation was effectively directed by Newton himself. The report found that Leibniz had concealed his knowledge of Newton's work — based on facts now known to be false. In response, Leibniz accused Newton and his followers of stealing Leibniz's own calculus and making errors in their applications of it. The dispute went on well after Leibniz's death in 1716, full of accusations and counter-accusations.
Nobody came out of the dispute well. Both Newton and Leibniz were capable of incredible mathematical discoveries, but their dispute demonstrated they were also capable of some rather less impressive behaviour.