Differentiation

Differentiation is a process in calculus that associates to a function f its derivative, which encodes the instantaneous rate of change of f with respect to its variable. Geometrically, the derivative at a point x is the slope of the tangent line to the graph y = f(x). It is defined by the limit f′(x) = lim_{h→0} (f(x+h) − f(x))/h, provided the limit exists. The derivative measures how f responds to small changes in x and forms the foundation for many analytical techniques and applications.

Notation and rules: Derivatives are written as f′(x) or dy/dx in single-variable contexts. For functions of several variables, partial derivatives ∂f/∂x, ∂f/∂y, and so on are used. Core rules include the power rule, constant multiple rule, sum rule, product rule, quotient rule, and chain rule. The chain rule states that if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x). These rules enable differentiation of a wide range of functions built from elementary functions.

Multivariable and geometric interpretation: For functions of several variables, the gradient ∇f collects all partial derivatives and points in the direction of steepest ascent. The differential provides a linear approximation: f(x + h) ≈ f(x) + ∇f(x) · h. The Fundamental Theorem of Calculus links differentiation and integration, while Taylor’s theorem gives higher-order approximations of f near a point.

Applications and properties: Differentiation is central in physics, engineering, economics, and beyond, appearing in optimization, motion analysis, and modeling. Differentiability on an interval implies continuity there, but continuity alone does not guarantee differentiability. Topics such as higher-order derivatives, implicit differentiation, and techniques like logarithmic differentiation are commonly used in practice.