afleiðu

Afleiðu is the Icelandic term for the derivative, a central concept in calculus describing how a function changes at a given point. For a real-valued function f defined near a, the derivative at a is f′(a) = lim h→0 (f(a+h) − f(a))/h, provided the limit exists. If it exists for all a in an interval, f′ is a function called the derivative of f on that interval.

Geometrically, the derivative at a point equals the slope of the tangent line to the graph of f at that point. It also represents instantaneous rate of change, such as velocity as the rate of change of position with respect to time.

Notationally, derivatives can be written as f′(x), df/dx, or Df(x). The process of finding derivatives is called differentiation.

Rules of differentiation include linearity and standard rules: the power rule (d/dx x^n = n x^(n−1)); the constant multiple rule (d/dx [c g(x)] = c g′(x)); the sum rule; the product rule (d/dx [u v] = u′ v + u v′); the chain rule (d/dx f(g(x)) = f′(g(x)) g′(x)). Common elementary derivatives include (x^2)' = 2x, sin x' = cos x, e^x' = e^x.

Applications span physics, engineering, economics, and beyond, including velocity and acceleration, optimization, and rate-of-change analysis.

Historically, the derivative emerged from the development of differential and integral calculus in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, with the terminology and notation evolving over time.