Differential calculus emerged during the pursuit of understanding motion and change. One of the central problems that spurred its development was related to the concept of instantaneous velocity. Philosophers and scientists, including ancient thinkers like Zeno, grappled with the idea of how to quantify motion and change over time.
In the late 17th century, mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz independently formulated the principles of calculus. They were driven by questions such as: How can we accurately determine the slope of a curve at a certain point? How can we understand the rate of change of a quantity with respect to another?
Newton, in particular, was interested in understanding the motion of planets and objects under the influence of gravity. He needed a way to describe how objects accelerated and changed velocity, which led him to develop the fundamental ideas of derivatives, which are central to differential calculus.
Overall, the physical problem of measuring change in motion resulted in profound mathematical development, allowing for the formulation of rules that can describe a wide range of physical phenomena.