Prologue and Historical Context
Key Questions

The calculus of singlevariable functions was developed around the same time by two different mathematicians; Gottfried Wilhelm Leibniz and Sir Isaac Newton.
Newton was a very independent and solitary man, and he rarely published any of his works, save for a few select manuscripts which he shared with only a handful of colleagues. In fact; the actual process for his "method of fluxions" as it was known at the time was not actually published until 1736, nine years after his death. Leibniz on the other hand contributed heavily to the academic world through the Berlin Academy and so his calculus caught on more quickly throughout Europe.
At the time, the question of who truly invented calculus was a matter of very heated debate and controversy, perhaps with some nationalistic undertones. Leibniz was often accused of plagiarizing works from Newton, and vice versa. Today most historians agree that both men deserve equal credit for independently advancing the same field.

Sir Isaac Newton was already well known for his theories of gravitation, and the motion of planets. His developments in calculus were to find a way to unify mathematics and the physics of planetary movement and gravity. He also introduced the notion of the product rule, the chain rule, Taylor series, and derivatives higher than the first derivative.
Newton mainly worked with function notation, such as:
#f(x)# to denote a function#f'(x)# to denote the derivative of a function#F(x)# to denote an antiderivative of a function
So, for example, the product rule looks like this:
#"Let "h(x)=f(x)g(x).#
#"Then " h'(x) = f'(x)g(x) + f(x)g'(x)# This notation can be confusing for some people, which is where Leibniz comes into the picture.

Gottfried Wilhelm Leibniz was a mathematician and philosopher. Many of his contributions to the world of mathematics were in the form of philosophy and logic, but he is much more well known for discovering the unity between an integral and the area of a graph. He was primarily focused on bringing calculus into one system and inventing notation that would unambiguously define calculus. He also discovered notions such as higher derivatives, and analysed the product and chain rules in depth.
Leibniz mainly worked with his own invented notation, such as:
#y=x# to denote a function, in this case, f(x) is the same as y#dy/dx# to denote the derivative of a function#intydx# to denote an antiderivative of a function
So, for example, the product rule looks like this:
#"Let "y=uv,# where u and v are both functions
#"Then " dy/dx = u(dv)/dx + v(du)/dx# This notation can be overwhelming for some people, which is where Newton comes into the picture.

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.