Differential operators are mathematical tools used to analyze functions by measuring how they change. They operate on functions to produce new functions, often representing rates of change, such as the derivative which indicates how a function's value changes with respect to its input. In calculus, the most common differential operator is the n-th derivative, which applies the differentiation process multiple times. Differential operators are essential in various fields, including physics, engineering, and economics, as they help model dynamic systems and solve differential equations that describe real-world phenomena.