Derivaatat

Derivaatat are mathematical concepts in calculus that describe the instantaneous rate of change of a function with respect to a variable. For a real-valued function f of a real variable x, the derivative at a point x0 is defined as the limit as h approaches 0 of [f(x0+h) − f(x0)]/h, provided the limit exists. The derivative is denoted f′(x) or df/dx, and it represents the slope of the tangent line to the graph of f at x0.

More generally, derivaatat exist for vector-valued functions; the derivative at a point is a linear map called the differential. For functions f: R^n → R, the gradient ∇f is the vector of partial derivatives. For f: R^n → R^m, the derivative is represented by the Jacobian matrix, which encodes how each component of f changes with respect to changes in the input variables.

Higher-order derivatives include the second derivative f″(x), which provides curvature information; the Hessian is the matrix of second partial derivatives. Directional derivatives measure the rate of change in a specified direction. The chain rule relates derivatives of composed functions; standard rules also include the product rule and the quotient rule.

Differentiability implies local linear approximation, and differentiability in turn implies continuity, though continuity does not guarantee differentiability. Applications of derivaatat span many fields, including physics (for example, velocity as the derivative of position), economics (marginal rates), biology, and engineering. In multivariable calculus, derivatives underpin optimization, motion, and dynamics. The Fundamental Theorem of Calculus connects derivatives with definite integrals, showing that differentiation and integration are inverse processes under suitable conditions.