In the mid 1600s, there already existed a few ideas on how to measure volumes, areas, and rates of change. None of them were very rigorous, and all of them were rather disjointed. We could calculate the volume of a sphere, the surface area of a cube, and the acceleration of a runner, but there was no single mathematical system that unified the three ideas.
The closest that we came for a long time was in the mid 1600s. Bonaventura Cavalieri worked on computing areas and volumes, but his methods were a little bit askew. Early versions of what we now know as calculus were worked on by mathematicians such as Pierre de Fermat, and proofs were written for some fundamental ideas (including the idea that a definite integral can be computed using a function's antiderivative). Calculus was still in its infancy, though, and more ideas still had to be proven and squeezed out of the mathematical world.
Enter Isaac Newton and Gottfried Leibniz. The two mathematicians independently developed notation and ideas for unifying those concepts, as well as prove other fundamental concepts of calculus. Newton focused on introducing techniques such as the product and chain rule, as well as applying calculus to motion (especially to astronomical bodies). Leibniz focused on unifying the ideas under one notation and gave calculus its name. Newton had his ideas first, but Leibniz published first; today, both mathematicians get the credit.
Since then there have been remarkably few adaptations to the grassroots ideas of calculus, only additions and applications.