Final answer:
Option (C), Newton and Leibniz both contributed to the development of calculus, which unites differential and integral calculus through its fundamental theorem. This branch of mathematics is critical for understanding and solving problems related to change and motion in various fields.
Step-by-step explanation:
The branch of mathematics that Isaac Newton and Gottfried Wilhelm Leibniz independently contributed to with the fundamental theorem that unites its two main types is calculus. This fundamental theorem is often considered the central doctrine of calculus. It connects two conceptually different parts of the subject: differential calculus and integral calculus. The power of calculus lies in its ability to deal with changes, which is critical in numerous fields including engineering, physics, economics, and more.
Calculus is essential for solving problems involving change and motion, and it is based on understanding limits, derivatives, integrals, and infinite series. Newton's work in mechanics, as noted in Philosophiae Naturalis Principia Mathematica, and his invention of calculus are profoundly influential in the understanding of the physical world and its mathematical description.