Derivates

A derivative is a fundamental concept in calculus that measures how 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: f'(x) = lim(h→0) [f(x+h) − f(x)] / h, provided this limit exists. The derivative, when it exists, is a function describing the instantaneous rate of change and, geometrically, the slope of the tangent line to the graph of f at a point.

Notation for the derivative includes f'(x), df/dx, and Df. For functions of several variables, partial derivatives ∂f/∂x describe rates of change along coordinate directions, while directional derivatives give rates of change in specified directions.

Common examples include f(x) = x^2, whose derivative is f'(x) = 2x, and f(x) = sin x, whose derivative is f'(x) = cos x. Derivatives obey several rules, including the power rule, product rule, quotient rule, and chain rule. Higher derivatives, such as f''(x), measure how the rate of change itself changes and are useful for studying concavity, inflection points, and optimization.

Existence and interpretation: differentiability is stronger than continuity; a function can be continuous but not differentiable at certain points (for example, |x| at x = 0). The derivative may fail to exist where a function has a cusp, a vertical tangent, a jump, or a discontinuity.

Applications are broad, spanning physics, engineering, and economics. Velocity is the derivative of position, acceleration is the derivative of velocity, and marginal analysis uses derivatives to study local behavior. The concept originated with Newton and Leibniz and remains central to differential calculus.