Derivative

A derivative in mathematics is a measure of how a function changes as its input changes. It quantifies the rate of change or slope of a function at a specific point and is fundamental in calculus. The derivative of a function \(f(x)\) at a point \(x\) is denoted as \(f'(x)\) or \(\frac{df}{dx}\), indicating the limit of the average rate of change over an infinitesimally small interval as that interval approaches zero.

Derivatives can be used to analyze the behavior of functions, such as identifying maximum and minimum points (local extrema), points of inflection, and increasing or decreasing intervals. They are also employed in various applications across physics, engineering, economics, and biological sciences to model systems that involve rates of change or dynamic interactions.

The process of finding a derivative is called differentiation. Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule, each governing how derivatives are computed for various types of functions. For example, the power rule states that the derivative of \(x^n\) (where \(n\) is a real number) is \(nx^{n-1}\).

There are different types of derivatives, including partial derivatives used in multivariable calculus to measure the change of a function with respect to one variable while holding others constant, and higher-order derivatives that analyze the curvature and concavity of functions.

Overall, derivatives are a core concept in mathematical analysis, providing essential tools for understanding dynamics and optimizing systems in scientific and engineering contexts.