Newton was born in 1642, Leibniz four years later. Calculus is a means for calculating the way quantities vary with each other, rather than just the quantities themselves. The bare bones of that idea had been hatching before either Newton or Leibniz was born. But they each wrote a full system of calculus. Newton began creating calculus around 1664-1666, but did not publish. Some people were aware of what he was doing, through letters and papers which Newton showed to people.
He did not publish until 1687 and later. Leibniz first studied calculus around 1672-1676, and published in 1684 and 1686. The ideas were similar, but the notation was different. And many of the lesser discoveries were different. Unaware that Newton was reported to have discovered similar methods, Leibniz realized that he was onto something “big”; he just didn’t realize that Newton was on to the same big discovery because Newton was remaining somewhat quiet about his breakthroughs. In fact, it was actually the delayed publication of Newton’s findings that caused the entire problem.
Evidence also shows that Newton was the first to establish the general method called the "theory of fluxions" it was the first to state the fundamental theorem of calculus and was also the first to explore applications of both integration and differentiation in a single work. However, since Leibniz was the first to publish a dissertation on calculus, he was given the total credit for the discovery for a number of years. This later led, of course, to accusations of plagiarism being hurled relentlessly in the direction of Leibniz.
There was speculation that Leibniz may have gleaned some of his insights from two of Newton's manuscripts on fluxions, and that that is
