Calculus was developed independently by both Isaac Newton and Gottfried Leibniz during the later part of the 1600s. For Newton, calculus was primarily a tool he needed for explaining the motion of the planets. It would be difficult to say precisely how he developed his ideas because he was secretive about his methods, but it certainly grew out of his understanding of the laws of motion and acceleration.
Newton's development of calculus may have grown out of his need to explain the acceleration of falling bodies. He knew that the speed of a falling object increases by a tiny amount every split second that it falls. Average acceleration was easy to calculate, but there was no mathematical process available for describing the position or velocity of the object at any given point in time. Calculus gave him a way of describing not only derivative functions like the rate of change over time but also the curved motion caused by the force of gravity, which allowed him to explain the elliptical motion of planets as conic sections.
While Newton and Leibniz were both brilliant mathematicians and instrumental in the development of calculus, both men built their systems upon ideas that had been around since antiquity. Early Greek, Egyptian and Chinese mathematicians all attempted to find the area of a circle by using the converging areas of inscribed or circumscribed polygons, a method that anticipates modern calculus.