derivatiiv

Derivatiiv is a central concept in calculus describing the instantaneous rate of change of a function with respect to its variable. It is commonly denoted f′(x) or dy/dx and can be defined as the limit of the average rate of change as the interval shrinks: f′(x) = lim h→0 [f(x+h) − f(x)]/h, provided the limit exists.

Interpretation and use: the derivative at a point equals the slope of the tangent line to the function’s graph there, and it provides the best linear approximation to the function near that point. Consequently, f(x+h) ≈ f(x) + f′(x)h for small h. Higher-order derivatives, such as f′′(x), measure changes in the rate of change and relate to concavity and curvature; Taylor's theorem expresses a function as a power series based on its derivatives at a point.

Notational and contextual variants: derivative notation includes f′, Df, and dy/dx. In multivariable calculus, the gradient ∇f and the Jacobian generalize the notion of a derivative to several variables or vector-valued functions. Differentiability at a point implies continuity there, but a function can be continuous without being differentiable (for example, |x| at x = 0).

Rules and examples: standard rules include the product, quotient, and chain rules. Examples: d/dx(x²) = 2x, d/dx(sin x) = cos x, d/dx(eˣ) = eˣ.

Applications: derivatives model motion (velocity from position), optimize objectives in economics, and appear in physics, engineering, and numerical methods. Related concepts include differentiability classes (C¹, C∞) and the study of analytic functions.