derivatif

Derivatif, or derivative, is a central concept in calculus describing the instantaneous rate of change of a function with respect to a variable. For a real-valued function f defined on an interval, the derivative at x0 is the limit as h approaches 0 of [f(x0+h) − f(x0)]/h, if the limit exists. The derivative is denoted f′(x0) or df/dx. For functions of several variables, partial derivatives ∂f/∂x_i measure the rate of change with respect to each coordinate, and the gradient ∇f aggregates these components.

The derivative has several interpretations: the slope of the tangent line to the graph of f at x0, the instantaneous rate of change, and the best linear approximation to f near x0. Differentiability at a point implies continuity there; a function can be continuous yet non-differentiable at a point.

Key rules include linearity, the product rule, quotient rule, and the chain rule. Examples: d/dx x^n = n x^{n−1}, d/dx sin x = cos x, d/dx e^x = e^x, d/dx ln x = 1/x. The chain rule yields d/dx f(g(x)) = f′(g(x))·g′(x). Higher-order derivatives (f′′, f′′′, …) describe how the rate of change itself changes.

The fundamental theorem of calculus links differentiation and integration: under suitable conditions, the integral of f′ over an interval equals f(b) − f(a). Derivatives are central in applications across physics, economics, biology, and engineering, where they describe velocity, marginal analysis, growth rates, and optimization. In multivariable settings, the gradient and directional derivatives underpin optimization and sensitivity analysis.