Dérivé

Dérivé, in calculus, is the derivative of a function. The dérivé at a point x of a function f is the limit of the average rate of change: lim_{h→0} [f(x+h) − f(x)] / h, provided the limit exists. If the limit exists, f is differentiable at x and the dérivé is denoted f′(x) or df/dx. For a function of several variables, partial dérivés ∂f/∂x_i generalize the concept, and the gradient collects these partial dérivés.

Geometrically, the dérivé is the slope of the tangent line to the graph of f at x, representing the instantaneous rate of change of f with respect to x. It is the foundation for many rules of differentiation, such as linearity, the power rule, the product rule, the quotient rule, and the chain rule. If a function is differentiable at a point, it is continuous there, but continuity does not in general imply differentiability.

Examples: d/dx x^n = n x^{n−1}; d/dx sin x = cos x (in radians); d/dx e^x = e^x; d/dx ln x = 1/x for x > 0. Higher-order derivatives are successive dérivés, denoted f″(x), etc. Applications span physics (velocity as the dérivé of position), engineering, economics (marginal analysis), and optimization.

Historically, the concept emerged in the 17th century via Newton and Leibniz. The term dérivé is the French form of derivative. See also differential, gradient, Jacobian.