What did Newton and Leibniz actually discover? Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a limit with its $\epsilon$-$\delta$ definition) that seem to have been developed only much later (in this case in the late $18$th and early $19$th century).
Hence the question - what is it that Newton and Leibniz discovered?  

This is actually quite a complicated question, since it spans two whole careers.
Some say calculus was not discovered by Newton and Leibniz because Archimedes and others did it first.  That's a somewhat simple-minded view.  Archimedes solved a whole slew of problems that would now be done by integral calculus, and his methods had things in common with what's now taught in calculus ("now" = since about 300 years ago), but his concepts were in a number of ways different, and I don't think he had anything like the "fundamental theorem".
I'm fairly sure Leibniz introduced the "Leibniz" notation, in which $dy$ and $dx$ are corresponding infinitely small increments of $y$ and $x$, and the integral notation $\int f(x)\,dx$.  I suspect Newton and Leibniz were the first to systematically exploit the fundamental theorem.  And the word "systematic" is also important here: Newton and Leibniz made the computation of derivatives and integrals systematic.
