partiellderivater

Partiellderivater is a term commonly used in advanced mathematics and engineering to denote the process of computing the rate of change of a multivariable function with respect to one of its variables while keeping the other variables constant. This concept is fundamental in calculus, forming the basis for the analysis of functions of several real or complex variables.

The partial derivative of a function f(x1, x2, …, xn) with respect to a variable xi is denoted ∂f/∂xi or del f/∂xi. Conceptually, it measures how f changes when xi is varied infinitesimally, holding xj (j ≠ i) fixed. Formally, if f is defined in a neighbourhood of a point (a1, …, an), then ∂f/∂xi (a1,…,an) = lim h→0 [f(a1,…,ai + h,…,an) − f(a1,…,ai,…,an)] / h, provided the limit exists.

For instance, for f(x, y) = x^2 y + sin(x y), the partial derivatives are ∂f/∂x = 2xy + y cos(xy) and ∂f/∂y = x^2 + x cos(xy). These derivatives are key tools in optimization, providing necessary conditions for local extrema in multivariate contexts through the vanishing of all partial derivatives.

Partial derivatives appear extensively in physics, where they describe how physical quantities vary across space and time, and in economics, where they evaluate marginal effects of variables. They also underpin advanced topics such as gradients, divergence, curl, and the Jacobian matrix. The notation and conceptual framework were first developed in the 17th and 18th centuries by mathematicians like Leibniz and Lagrange, and have since become central to differential calculus.