Both Newton and Leibniz are credited as the inventors of calculus.
While Leibnitz's interest in the subject was a mixture of philosophy and pure mathematics, it was Newton who recognized the usefulness of it in natural sciences.
The key year was 1684.
In Germany, in 1675, Leibniz applied integral calculus for the first time to find the area under the graph of a function. He did not publish anything about calculus until 1684.
In early 1684, in Britain, Christopher Wren, Robert Hooke, and Edmond Halley (the comet guy) had a discussion about how to derive Keppler's observed laws of planetary motion from physical laws. Halley asked Newton for help. The question was "what would be the shape of planet orbit if attraction force follows the inverse-square law?". "Ellipse", answered Newton, who had already finished the calculations. In late 1684, Newton sent Halley a 9-page manuscript prepared for print. This would become the core of his "Principia", published in 1687.
We still see the legacy of Leibniz and Newton in notation. Leibnitz invented symbols ∫ and dy/dx, while Newton used notation similar to f '(x) to designate derivative.
It should be noted those were only crude sketches of calculus we learn today. Many things they assumed to be correct had to wait for strict formal proof. That work of putting calculus on the firm formal ground was mostly completed by about 1850. Those branches you mentioned were contributed during this period by great mathematicians like Bernoulli brothers, Euler, Cauchy, Gauss, Riemann, Weierstrass, and many others.