Differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, which studies the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differential calculus has applications in nearly all quantitative disciplines, including physics, where the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body, which leads to the famous F = ma equation associated with Newtons second law of motion. Differential calculus is also used in economics, biology, engineering, and many other fields.
The differential of a function is a concept in calculus that represents the principal part of the change in a function with respect to changes in the independent variable. It is defined as the product of the derivative of the function and an infinitesimal change in the independent variable. The differential is used to approximate the change in the function for small changes in the independent variable. The derivative is the fundamental tool of differential calculus and is used to show the rate of change of a function. It measures the steepness of the graph of a function and defines the ratio of the change in the value of a function to the change in the independent variable.
