Derivatives

In mathematics, a derivative measures the instantaneous rate at which a function changes with respect to a variable. For a function f of a real variable x, the derivative is denoted f'(x) or df/dx and is defined as the limit as h approaches 0 of [f(x+h) − f(x)]/h, provided the limit exists.

Geometrically, the derivative at a point is the slope of the tangent line to the graph of f at that point. Derivatives describe rates of change, such as velocity being the rate of change of position with respect to time.

Fundamental rules include the power rule, the product rule, the quotient rule, and the chain rule. Higher-order derivatives are derivatives of derivatives; the second derivative informs about concavity and can indicate acceleration in a physical context. Partial derivatives extend the notion to functions of several variables.

Derivatives are central to optimization and analysis: setting f'(x) = 0 identifies candidate extrema, and the second derivative test or higher-order tests determine their nature. They also support curve sketching, differential equations, and the study of motion and change.

In finance, derivatives are contracts whose value derives from an underlying asset, such as stocks, bonds, commodities, or indices. Common types include futures, options, forwards, and swaps. They are used for hedging risk, speculation, and arbitrage, and are priced using models that incorporate factors such as volatility and interest rates.