derivatív

Derivatív is the Hungarian term for derivative, used in mathematics and finance. In calculus, the derivative of a function f at a point x measures the instantaneous rate of change of f with respect to its input. It is defined as the limit f'(x) = lim_{h→0} (f(x+h) - f(x))/h, when this limit exists. The derivative can be interpreted as the slope of the tangent line and as the best linear approximation of f near x.

Rules of differentiation: If f is differentiable, its derivative can be computed using rules such as the chain rule, product rule, and quotient rule. For functions of several variables, partial derivatives ∂f/∂xi describe rates of change along coordinate directions, and the gradient vector collects these partials. Higher-order derivatives include the second derivative f''(x) and mixed partials.

Geometric and analytic use: The derivative provides local linearization via f(x+h) ≈ f(x) + f'(x)h. The second derivative informs about concavity and inflection. In multiple dimensions, the Hessian matrix contains second partial derivatives, and the Jacobian matrix describes derivatives of vector-valued functions.

Applications: Derivatives are central in physics for velocity and acceleration, in economics for marginal analysis, and in optimization for extrema. They underpin numerical methods and dynamical-systems studies. The concept has historical roots in the work of Newton and Leibniz, who developed calculus to formalize instantaneous change.

Financial derivatives: The term denotes instruments whose value depends on underlying assets such as stocks, bonds, commodities, or indices. Common types include futures, forwards, options, and swaps. They are used for hedging, arbitrage, and speculative trading. Pricing relies on models linking payoff to the underlying state, with risk measured by sensitivities called the Greeks (delta, gamma, theta, vega, rho).

Computation and theory: Derivatives can be obtained symbolically, numerically (finite differences), or by automatic differentiation in software, enabling precise differentiation of complex models and simulations.