derivatan

Derivatan, in the mathematical sense often called the derivative, is a measure of how a function changes as its input changes. For a real-valued function f defined near a point x, the derivatan at x is f′(x) = lim h→0 [f(x+h) − f(x)] / h, if this limit exists. This limit, when it exists, represents the instantaneous rate of change of f with respect to its argument and the slope of the tangent line to the graph of f at x.

Notation and interpretation vary. Common notations include f′(x), df/dx, and Df(x). For functions of several variables, partial derivatives ∂f/∂x describe the rate of change in one variable while holding others constant, and the gradient vector collects all partial derivatives. The derivatan provides a local linear approximation: near a, f(x) ≈ f(a) + f′(a)(x−a).

Basic rules and examples help compute derivatan. If f(x) = x^n, then f′(x) = nx^{n−1}. The derivative of sin x is cos x, and the derivative of a constant is zero. Higher-order derivatives are successive derivatives, with the second derivative f″(x) describing the curvature. Key rules include the power rule, product rule, quotient rule, and chain rule, enabling differentiation of a wide range of functions.

Applications are widespread. In physics, velocity is the derivatan of position with respect to time, and acceleration is the second derivatan. In economics, the marginal rate of change is modeled by derivatives. The concept underpins many areas of mathematics and applied sciences, and its development is historically linked to the work of early contributors such as Leibniz and Newton. In Indonesian and some other languages, the term for derivative is derivatan.