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Differential calculus, often referred to as differentials, is a branch of calculus that deals with rates of change and slopes of curves. It is concerned with the study of how quantities change in relation to each other. The fundamental concept in differential calculus is the derivative, which represents the rate at which a quantity is changing at a specific point. This is typically denoted as f'(x) or dy/dx, where y is a function of x.

The derivative of a function at a point is defined as the limit of the difference quotient as the change in the input approaches zero. This limit gives the slope of the tangent line to the curve at that point. The derivative has numerous applications in mathematics, physics, engineering, and economics. It is used to find the velocity and acceleration of moving objects, the rate of change of functions, and to optimize functions.

One of the key rules in differential calculus is the chain rule, which allows for the differentiation of composite functions. Other important concepts include the product rule, quotient rule, and the derivatives of basic functions such as polynomials, exponentials, and logarithms. The Fundamental Theorem of Calculus links differential calculus with integral calculus, stating that differentiation is the reverse process of integration.

Differential calculus is essential for understanding the behavior of functions and their graphs. It provides tools for analyzing the local properties of functions, such as determining where a function is increasing or decreasing, finding critical points, and understanding concavity. These concepts are crucial in various fields, including science, engineering, and economics, where understanding rates of change and optimization is vital.