Newton's Calculus: What He Came Up With & Notation Used

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Newton developed foundational concepts in calculus, focusing on limits and real-world applications, but his notation differed significantly from what is used today. Most modern calculus notation, including symbols for derivatives and integrals, was introduced by Leibniz, who favored a more abstract approach. While Newton contributed to the understanding of derivatives and integrals, he did not formalize them in the same way as Leibniz. The credit for the invention of calculus is often shared between the two due to their distinct methodologies and contexts. Ultimately, Newton's contributions laid essential groundwork, even if his notation was not adopted in contemporary calculus.
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How much calculus did Newton himself actually come up with? And what did it look like? I've read that most of the notation we use today was actually invented by Leibniz- if that's true, then what did Newton's notation look like?
 
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Netwon went a long way and for this reason 'credit' for the 'invention' of calculus is often shared as Newton was based primarily on true reality nad the use of limits where as Leibniz was far mroe embracing of the abstract and infinite.

Perhaps this reflects the very different way they approached the problem due to their situations.
 
So did Newton come up with the equations for equations for derivatives and integrals that we use today? I'm thinking of things like (d/dx)(sinx) = cosx, and such.
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...

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