Tuletiste

Tuletiste is an Estonian term with two related uses, connecting language and mathematics. In linguistic terms, tuletiste is the genitive plural form of the noun tuletis, which means derivative. It appears in sentences discussing derivatives or rates of change, for example when describing the values or properties of derivatives.

In mathematics, a derivative (tuletis in Estonian) describes the instantaneous rate at which a function changes as its input changes. It is defined as the limit of the average rate of change as the input increment approaches zero. The derivative at a point is usually denoted f′(x) or df/dx and can be interpreted geometrically as the slope of the tangent line to the graph of the function at that point. Derivatives follow standard rules, including the power rule, product rule, and chain rule, which allow the differentiation of a wide class of functions.

Common examples include d/dx(x^2) = 2x and d/dx(sin x) = cos x. Higher derivatives measure how the rate of change itself changes; for instance, d^2/dx^2(x^2) = 2. In multivariable calculus, derivatives extend to partial derivatives, gradients, and Hessians, providing tools to analyze functions of several variables and to study optimization and curvature.

Derivatives have broad applications across science and engineering. In physics, velocity is the derivative of position with respect to time; in economics, marginal changes are described by derivatives; in biology and engineering, they underpin models of growth, motion, and response to stimuli. The concept is foundational to differential calculus and to many numerical methods used to approximate solutions to equations and optimization problems.