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Derivatives in mathematics refer to the fundamental concept of calculus that describes how a function changes as its input varies. Specifically, the derivative of a function at a given point represents the rate at which the function's output value changes with respect to changes in its input value. This is often interpreted as the slope of the tangent line to the function's graph at that point.

The formal definition of a derivative involves the limit of the difference quotient as the interval between two points approaches zero. For a function *f(x)*, the derivative *f'(x)* is defined as:

f'(x) = lim(h→0) [f(x + h) - f(x)] / h

This concept is widely used across various fields, including physics, engineering, economics, and computer science, to model dynamic systems and optimize processes. For example, in physics, derivatives describe velocity (the rate of change of position) and acceleration (the rate of change of velocity). In economics, marginal cost and marginal revenue are derivatives of cost and revenue functions, respectively.

Derivatives can be computed for a wide range of functions, including polynomials, trigonometric functions, exponential functions, and more complex composite functions. Rules such as the power rule, product rule, quotient rule, and chain rule simplify the process of differentiation for specific types of functions. The power rule states that if *f(x) = x^n*, then *f'(x) = n*x^(n-1)*. The chain rule is used for composite functions, allowing differentiation of functions within functions.

Higher-order derivatives represent the rate of change of derivatives themselves. For instance, the second derivative of position with respect to time gives acceleration. Derivatives are also foundational in advanced topics such as integration, differential equations, and multivariable calculus. Understanding derivatives is essential for analyzing functions, solving optimization problems, and modeling real-world phenomena.