Partial DerivativeEdit

A partial derivative is a basic construct in calculus that measures how a function of several variables changes when we vary one variable at a time, holding the others fixed. If f is a function from a multidimensional domain to the real numbers, the partial derivative with respect to the i-th coordinate x_i describes the instantaneous rate of change of f along the x_i direction. This concept is central to understanding how complex systems respond to local changes, whether in physics, economics, engineering, or data analysis.

In formal terms, for a function f: R^n → R, the partial derivative with respect to x_i is written as ∂f/∂x_i. It is defined by the limit ∂f/∂x_i = lim_{h→0} [f(x_1, ..., x_i + h, ..., x_n) − f(x_1, ..., x_i, ..., x_n)] / h, provided this limit exists. The collection of all first-order partial derivatives forms the gradient, denoted ∇f or grad f, which points in the direction of steepest ascent of f. The gradient is a vector of the form [∂f/∂x_1, ∂f/∂x_2, ..., ∂f/∂x_n], and it plays a key role in optimization and in characterizing local behavior of f. For a given function, studying the gradient and how it changes across the domain connects to broader ideas in multivariable calculus and differentiability.

Basics - Definition and notation: The partial derivative with respect to a coordinate measures sensitivity of f to changes along that coordinate while keeping other coordinates fixed. See Function and Gradient for related concepts. - Simple example: Let f(x,y) = x^2 y + sin(y). Then ∂f/∂x = 2xy and ∂f/∂y = x^2 + cos(y). This illustrates that different coordinates can contribute to the rate of change in different ways. - Existence and interpretation: The mere existence of a partial derivative at a point does not guarantee continuity or differentiability of f at that point, but it does give a local linear approximation in the corresponding coordinate direction. See Continuity and Differentiability for related ideas.

Geometric interpretation and the gradient - The gradient ∇f encodes the direction and rate of fastest increase of f. Its magnitude equals the rate of steepest ascent, and its negative direction points toward the steepest descent. The gradient is a central tool in optimization, constrained or unconstrained, and connects to Jacobian matrix and Hessian matrix in higher-order analysis. - Level sets and surfaces: Partial derivatives help describe how level sets (where f is constant) bend and tilt in space. The local behavior near a point is captured by a linear approximation using the first-order terms in a Taylor expansion.

Higher-order derivatives and mixed partials - Second-order partial derivatives measure curvature. The Hessian matrix H of f collects the second-order partial derivatives, with entries ∂^2f/∂x_i∂x_j. Analyzing H informs about convexity, local minima and maxima, and the geometry of the graph of f. - Mixed partial derivatives: The order of differentiation can matter in non-smooth situations. Under mild regularity conditions (e.g., if the second-order partials are continuous in a neighborhood), Schwarz's theorem (also called Clairaut’s theorem) ensures ∂^2f/∂x_i∂x_j = ∂^2f/∂x_j∂x_i. See Schwarz's theorem for details.

Rules, chain rule, and differentiation under the integral sign - Linearity and basic rules: Partial derivatives satisfy linearity in each argument, and differentiation with respect to one coordinate commutes with addition and scalar multiplication of functions in the appropriate conditions. - Chain rule: When f depends on several intermediate variables that themselves depend on a parameter or on another set of coordinates, the chain rule yields expressions for the overall rate of change in terms of the partial derivatives of f and the derivatives of the inner functions. See Chain rule and Implicit differentiation for related tools. - Total derivative concept: The total derivative generalizes the idea of a single-variable derivative to multivariable contexts, accounting for all ways the input variables can change. See Total derivative for a broader view.

Applications and examples - Optimization and economics: Partial derivatives enable marginal analysis, sensitivity studies, and optimization under constraints. Economists often study how small changes in input prices or resource levels affect outcomes modeled by f. See Lagrange multipliers for a standard technique in constrained optimization. - Physics and engineering: Many physical fields (temperature, pressure, potential) are modeled as functions of space and time. Partial derivatives describe local rates of change, flux, and diffusion processes, with links to equations like the Heat equation and other partial differential equations. - Data science and engineering: In multivariate data, partial derivatives appear in sensitivity analyses, gradient-based optimization algorithms, and in defining local approximations used by algorithms that learn from data. See Differentiable function and Optimization.

Computation and numerical aspects - Analytical computation: For many standard functions, partial derivatives can be computed symbolically using rules of differentiation. See Symbolic differentiation and Differentiation for methods. - Numerical differentiation: When an explicit form is unavailable or too complex, finite-difference methods approximate partial derivatives by evaluating f at nearby points. See Finite difference and Numerical differentiation for common schemes and error considerations. - Software and tools: Computer algebra systems and numerical libraries implement partial differentiation as part of broader functionality in Mathematics software and Computational science ecosystems.

See also - Gradient - Jacobian matrix - Hessian matrix - Chain rule - Total derivative - Implicit differentiation - Optimization - Multivariable calculus