derivace

Derivace, in mathematical usage often equated with the derivative, is the concept of the instantaneous rate of change of a function with respect to its variable. For a real-valued function f defined on an interval, the derivative at a point x0 is defined as f'(x0) = lim_{h→0} (f(x0 + h) − f(x0)) / h, provided the limit exists. Geometrically, the derivative represents the slope of the tangent line to the graph of f at x0 and thus characterizes how the function value changes in small neighborhoods around x0.

Notationally, derivatives are written as f'(x), df/dx, or using differential notation df; partial derivatives in multivariable contexts are denoted ∂f/∂xj. Fundamental rules govern differentiation, including linearity (d/dx [af + bg] = a f' + b g'), the product rule (fg)' = f'g + fg', the quotient rule, and the chain rule for composed functions. The derivative provides a local linear approximation: f(x) ≈ f(x0) + f'(x0)(x − x0) when x is near x0.

In several variables, derivatives extend to partial derivatives, the gradient ∇f, the Jacobian matrix for vector-valued functions, and the Hessian for second-order information. Differentiation is central in physics, where velocity is the time derivative of position and acceleration is the second derivative; in optimization, derivatives identify critical points where f'(x) = 0.

Differentiability at a point implies continuity there, though a function can be continuous but not differentiable at some points. Derivace thus names a foundational tool in calculus and analysis, with wide-ranging theoretical and applied uses.