Why are there two separate branches of calculus?

Historically, there were two branches, differential and integral, right? Did they only form a unified calculus after the discovery of the Fundamental Theorem of Calculus?

Today, are the two branches still researched independently of each other?

• In a sense, yes, differentiation and integration are still studied separately. Modern integrals tend to involve more measure theory (most commonly, the Lebesgue integral). Certain Banach Spaces allow for a "Radon-Nikodym derivative" of one measure with respect to the other, and is defined purely in terms of the integral. Compare and contrast to an indefinite (Riemann) integral, which is defined purely in terms of the derivative! – Theo Bendit Mar 28 '18 at 5:27
• By mid-20th century it was clear that both the processes of integration and differentiation were (very important) examples of continuous linear maps (from one topological vector space of functions to a possibly different topological vector space of functions. Although some parts of the standard curriculum preserve an alleged separation between "integration" and "differentiation", this is mostly due to inertia rather than scientific content. – paul garrett Mar 28 '18 at 13:03

Calculus just means a way of calculating. There are many types of calculus, some of which are modern and currently actively studied (e.g. functional calculus, umbral calculus, difference calculus), while some are more or less fully understood. Differential and Integral calculus, i.e. the standard "calculus sequence", falls in the latter category. Note: mathematical analysis is a very active field of study, but it consists of generalizations and applications of the calculus sequence.

There are also abandoned types of calculus, like techniques for calculating square roots by hand example, which used to be taught like long division in school.

It depends on what you're referring to when you use the term "calculus".

Calculus itself comes from the Latin word meaning small pebble used for calculation, and is used to not only refer to differential and integral calculus as you describe. There are many other fields that have the name "calculus" to describe methods of calculation, such as propositional calculus, lambda calculus, etc. Before "standard" calculus, the term was widely used to refer to any field of mathematics.

But in the scope of differential and integral calculus, you are correct; the idea of infinitesimal change and infinitesimal summation predate Newton and Leibniz by centuries. The Fundamental Theorem of Calculus was only developed in the 17th and 18th centuries to connect these ideas. First introduced by Leibniz's and Newton's precursors, it was refined by these two. "Standard" calculus was then rigorized by later figures with the definition of limits such as used in derivatives, contrary to "infinitesimal ratios" presented by Leibniz.

historically, there were two branches right? differential and integral? Did they only form a unified calculus after the discovery of Fundamental Theorem of Calculus?

Are the two branches today still researched independently of each other?

in general there are two branches, diff and integral, but these are not two separate branches, not really. The derivative and integral are inverses of each other, like multiplication and division. Calculus had two independent discoverers, Newton and Leibniz. Newton is given credit for being the first to discover calculus. With Newton integral calculus came first, what he called fluxions, and differential came later. Newton used fluxions (what we call the fundamental theorem) to calculate the area under curves. Fluxions was initially derived from the (Newton's) binomial theorem. Leibniz, however, discovered differential calculus before integral, as best we know, as his treatise on differential calculus was published some time before his treatise on integral calculus. Newton is given credit for being the first to discover calculus, even though Leibniz published his findings before Newton did. Newton was very late in publishing, but his work predated Leibniz. We get our present day notation (dy/dx, etc) from Leibniz. We also get the integral sign from Leibniz. It comes from the Latin summa, meaning sum. Calculus itself is a Latin word meaning pebble, used for calculation in ancient times, as noted in a previous post.