Deriverte

Deriverte are the instantaneous rate of change of a function, and in Norwegian mathematics the term refers to the derivative. For a real-valued function f defined on an interval, the derivative at a point x0 is f'(x0) = lim_{h→0} (f(x0+h) − f(x0))/h, if this limit exists. It can be interpreted geometrically as the slope of the tangent line to the graph of f at x0 and as the instantaneous rate of change of f with respect to x.

Notation can vary: f' (prime), Df, or df/dx for a function of one variable. For functions of several variables, partial derivatives ∂f/∂xi measure the rate of change in a specific coordinate direction. Existence of the derivative implies continuity at the point, but a function can be continuous without being differentiable; common sources of non-differentiability include sharp corners and cusps.

Common differentiation rules include the power rule, sum rule, product rule, quotient rule, and chain rule. The derivative of common functions: d/dx x^n = n x^{n−1}, d/dx sin x = cos x, d/dx e^x = e^x. Higher derivatives measure changes in the derivative itself and provide information about concavity via the second derivative f''.

Derivatives have wide applications: physics links velocity to position, economics to marginal analysis, and optimization uses derivatives to locate stationary points. The concept originated independently in the work of Newton and Leibniz during the 17th century and has since become a central tool in mathematical analysis.