derivált

Derivált is the Hungarian term for the mathematical derivative. In calculus, the derivative of a function f with respect to a variable x measures the instantaneous rate of change of f as x varies. It is denoted f'(x) or df/dx, and the derivative at a point a is defined as the limit as h approaches 0 of [f(a+h) − f(a)]/h, provided the limit exists.

A derivative function f' is defined on those points where f is differentiable; at such points f' gives the slope of the tangent line to the graph of f at x. If a function is differentiable on an interval, it is continuous on that interval. Differentiation rules include the constant rule, the sum rule, the product rule (fg)' = f'g + fg', the quotient rule (f/g)' = (f'g − fg')/g^2, and the chain rule (f∘g)' = f'(g(x))·g'(x). Some functions have well-known derivatives: powers, exponentials, logarithms, and trigonometric functions. Higher-order derivatives are obtained by differentiating again, denoted f'', f''', and so on; the second derivative relates to concavity and inflection points.

Geometric interpretation and applications: the derivative at a point equals the slope of the tangent line, representing the instantaneous rate of change. Derivatives are central to optimization, physics, economics, biology, and numerical methods, including Newton’s method for finding roots.

History: the concept emerged in the 17th century during the development of calculus, with contributions from both Isaac Newton and Gottfried Wilhelm Leibniz. In Hungarian mathematical literature, the term derivált is used to denote this fundamental concept.