Tuletis

Tuletis, or derivative, is a central concept in calculus that quantifies the instantaneous rate of change of a function. In Finnish mathematics, tuletis refers to the derivative of a real-valued function of a real variable, and it is denoted by f'(x), dy/dx, or Df(x).

For a function f defined on an interval I around x0, the derivative at x0 is f'(x0) = lim_{h→0} [f(x0+h) - f(x0)]/h, if the limit exists. Example: f(x) = x^2 has derivative f'(x) = 2x, so f'(3) = 6.

Geometrically, the derivative at x0 is the slope of the tangent line to the curve y = f(x) there. It also represents the instantaneous rate of change of f with respect to x. If a function is differentiable at x0, it is continuous there; conversely, a function can be continuous but not differentiable (for example, |x| at x = 0).

Common rules include the chain rule, the product rule, and the quotient rule, which give derivatives of composite, product, and quotient functions. In several variables, partial derivatives form the gradient and Jacobian, and the derivative becomes a linear map described by the Jacobian matrix.

Higher-order derivatives measure further rates of change, with f''(x) indicating concavity and inflection points. Derivatives are widely used in physics, economics, biology, and engineering to model rates, optimize goals, and analyze motion.

Historically, the concept arose in the 17th century with Isaac Newton and Gottfried Wilhelm Leibniz. Leibniz introduced the dy/dx notation, while f'(x) or prime notation was popularized later by other mathematicians.