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
Common examples include f(x) = x^2, whose derivative is f'(x) = 2x, and f(x) = sin x, whose derivative
Existence and interpretation: differentiability is stronger than continuity; a function can be continuous but not differentiable
Applications are broad, spanning physics, engineering, and economics. Velocity is the derivative of position, acceleration is