Afleiða

Afleiða is the Icelandic term for the derivative in calculus. It denotes the instantaneous rate of change of a function with respect to its variable. For a function f of a real variable x, the afleiða at x is defined as the limit as h approaches 0 of (f(x+h) − f(x)) / h, provided the limit exists. This derivative is denoted f'(x) or df/dx, and equals the slope of the graph of f at x. The differential dy = f'(x) dx expresses small changes and underpins linear approximation.

Common differentiation rules include the power rule, product rule, and chain rule. Examples: d/dx x^n = n x^{n-1}; d/dx (uv) = u'v + uv'; d/dx f(g(x)) = f'(g(x)) g'(x). Derivatives can be taken with respect to more than one variable (partial derivatives) and lead to concepts such as gradients and Jacobians in multivariate calculus.

Higher-order derivatives, such as f''(x), describe how the rate of change itself changes and relate to concavity. The second derivative test helps determine local maxima and minima. Historically, the concept arose in the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Today afleiða is a fundamental tool across science, engineering, and economics, with widespread notation and applications in physics (velocity, acceleration), optimization, and modeling.