1 Definition
A partial derivative describes how a function of several variables changes when only one variable is varied and the others are kept fixed. It extends the idea of the ordinary derivative to settings with more than one input, such as functions of two or three coordinates. Partial derivatives are central to multivariable calculus and provide a basic way to measure directional change along coordinate axes.
1.1 Multivariable functions
A multivariable function assigns a value to each point in a domain with more than one coordinate. Common examples include functions of two variables, such as a surface height \(f(x,y)\), and functions of three variables, such as temperature \(T(x,y,z)\). These functions model quantities that depend on several inputs at once.
1.2 Holding variables constant
To compute a partial derivative, one variable is treated as the changing input while the remaining variables are regarded as constants. This isolates the effect of one coordinate direction. For example, in a function \(f(x,y)\), the partial derivative with respect to \(x\) measures change in \(f\) as \(x\) varies and \(y\) stays fixed.
1.3 Notation
Several notational systems are used for partial derivatives, and the choice often depends on the discipline or context. All of them indicate differentiation with respect to one variable in a multivariable setting.
1.3.1 Leibniz notation
Leibniz notation uses the symbol \(\partial\) to distinguish partial derivatives from ordinary derivatives. For a function \(f(x,y)\), the derivative with respect to \(x\) is written \(\frac{\partial f}{\partial x}\). This notation makes the differentiated variable explicit and is widely used in calculus and applied mathematics.
1.3.2 Subscript notation
Subscript notation records the variable of differentiation as a subscript, as in \(f_x\) or \(f_y\). This form is compact and common in textbooks, especially when many derivatives are discussed in sequence. Higher-order derivatives may be written as \(f_{xx}\), \(f_{xy}\), and similar expressions.
1.3.3 Operator notation
Operator notation emphasizes the differentiation operator itself, for example \(\partial_x f\). It is useful in theoretical work because it highlights partial differentiation as an operation acting on a function. In some contexts, the operator may be written as \(D_x\) or a related symbol.
2 Basic computation
Partial derivatives are usually computed by treating the chosen variable as the only variable and differentiating in the usual way. The other variables are regarded as fixed parameters. This makes the process similar to ordinary differentiation, but repeated separately for each input variable.
2.1 Direct differentiation
For an expression such as \(f(x,y)=x^2y+3y\), the partial derivative with respect to \(x\) is found by differentiating \(x^2y\) as if \(y\) were constant, giving \(2xy\). The partial derivative with respect to \(y\) treats \(x\) as constant, producing \(x^2+3\). Direct differentiation is the standard method for elementary formulas.
2.2 Partial derivatives of elementary functions
Elementary functions such as polynomials, exponentials, logarithms, and trigonometric expressions are differentiated using the same basic rules as in single-variable calculus. Products, powers, and compositions are handled one variable at a time. For instance, if \(f(x,y)=e^{xy}\), then \(\frac{\partial f}{\partial x}=ye^{xy}\) and \(\frac{\partial f}{\partial y}=xe^{xy}\).
2.3 Higher-order partial derivatives
Once a first partial derivative has been computed, it can itself be differentiated with respect to one or more variables. This produces second-order, third-order, and higher-order partial derivatives. These derivatives capture more detailed information about curvature and local variation.
2.3.1 Mixed partial derivatives
Mixed partial derivatives involve differentiating with respect to different variables in succession, such as \(\frac{\partial^2 f}{\partial y\,\partial x}\). They measure how the rate of change in one direction depends on another direction. In many smooth cases, the order of differentiation does not affect the result.
2.3.2 Clairaut's theorem
Clairaut's theorem states that, under suitable continuity conditions, mixed partial derivatives are equal when the order of differentiation is reversed. In practical terms, this means \(\frac{\partial^2 f}{\partial x\,\partial y}=\frac{\partial^2 f}{\partial y\,\partial x}\) for many well-behaved functions. The result is an important tool in analysis and geometry.
3 Geometric interpretation
Partial derivatives have a clear geometric meaning when a function is represented as a surface or as a scalar field. They describe local slopes, tangent behavior, and variation along coordinate directions. This interpretation connects algebraic formulas to shapes in space.
3.1 Rate of change along coordinate axes
The partial derivative with respect to one variable gives the slope of the function in the direction of that variable, with other coordinates fixed. For a surface \(z=f(x,y)\), \(\frac{\partial f}{\partial x}\) represents the slope of the curve formed by intersecting the surface with a plane parallel to the \(xz\)-plane. Similar reasoning applies to the \(y\)-direction.
3.2 Tangent planes and linear approximation
At a point on a smooth surface, partial derivatives determine the tangent plane. This plane provides the best linear approximation to the function near that point. In local analysis, the surface is often approximated by a linear expression built from its partial derivatives, which is useful for estimating small changes.
3.3 Level surfaces
A level surface is the set of points where a multivariable function has a constant value. Partial derivatives help describe how the function changes as one moves across these sets. They also relate to the geometry of level curves and surfaces by indicating directions of increase or decrease.
4 Rules and properties
Partial derivatives satisfy many of the familiar rules from ordinary differentiation. These rules make it possible to work with complicated expressions systematically. They also support more advanced techniques such as implicit differentiation and multivariable composition.
4.1 Sum and product rules
The sum rule states that the partial derivative of a sum is the sum of the partial derivatives. The product rule applies when two differentiable functions are multiplied, with the derivative distributed across the factors in the usual way. These rules are identical in form to their single-variable counterparts.
4.2 Quotient rule
When a function is written as a quotient, its partial derivative can be found using the quotient rule, provided the denominator is nonzero. The rule combines differentiation of the numerator and denominator in a structured formula. It is often used for rational expressions involving several variables.
4.3 Chain rule
The chain rule describes how partial derivatives behave under composition of functions. It is essential whenever one multivariable function depends on another through intermediate variables. This rule underlies much of applied calculus and many coordinate transformations.
4.3.1 Single-variable intermediate functions
If a multivariable function depends on a single parameter through several intermediate quantities, the derivative is found by combining the partial derivatives of the outer function with the derivatives of the inner quantities. This is the multivariable analogue of the ordinary chain rule. It is especially useful for parametrized curves and time-dependent systems.
4.3.2 Multivariable chain rule
When both the outer and inner functions have several variables, the chain rule is organized through sums of products of partial derivatives. Each path of dependence contributes to the final result. This formulation is often written compactly using matrices or differentials.
4.4 Implicit differentiation
Implicit differentiation is used when variables are related by an equation rather than written as an explicit function. Partial derivatives help determine how one variable changes in response to another while respecting the constraint. This method is common for surfaces and curves defined implicitly.
5 Differential operators
Partial derivatives are the building blocks of several important operators in multivariable calculus. These operators package derivative information in forms suited to geometry, physics, and systems of equations. They also express local behavior in concise notation.
5.1 Gradient
The gradient is the vector of partial derivatives of a scalar function. It points in the direction of greatest increase and has magnitude related to the steepness of that increase. Gradients are widely used in optimization and in the study of scalar fields.
5.2 Jacobian matrix
The Jacobian matrix collects first-order partial derivatives of a vector-valued function. Each entry records how one output component changes with respect to one input variable. It is a key tool for linearization, coordinate changes, and multivariable mappings.
5.3 Hessian matrix
The Hessian matrix is formed from second-order partial derivatives of a scalar function. It captures curvature and local convexity properties near a point. In optimization, it helps distinguish minima, maxima, and saddle points.
5.4 Divergence and curl
Divergence and curl are differential operators built from partial derivatives of vector fields. Divergence measures local expansion or contraction, while curl describes rotational tendency. These concepts are fundamental in vector calculus and continuum physics.
6 Applications
Partial derivatives appear in many scientific and technical fields because they describe change in systems with multiple inputs. They are used to study rates, optimize objectives, and formulate laws governing spatially varying quantities. Their flexibility makes them useful across theory and computation.
6.1 Optimization of several variables
In optimization, partial derivatives help locate points where a function reaches extreme values or changes behavior. By examining where derivatives vanish or fail to exist, one can identify candidates for optimal solutions. The second derivatives then help classify these candidates.
6.1.1 Critical points
Critical points are points where the first partial derivatives are zero or undefined. They often include local maxima, local minima, and saddle points. Further analysis, usually with the Hessian, is required to determine the type of each point.
6.1.2 Lagrange multipliers
Lagrange multipliers are used to optimize a function subject to a constraint. The method introduces an auxiliary variable so that the partial derivatives of the objective and constraint can be related systematically. It is a standard technique in constrained optimization.
6.2 Physics and engineering
In physics and engineering, partial derivatives describe how quantities such as temperature, pressure, stress, and electric potential vary across space and time. They are used in heat flow, fluid behavior, elasticity, and field theory. Many governing equations in these areas are written directly in terms of partial derivatives.
6.3 Economics and science
In economics, partial derivatives measure how output, cost, or utility changes when one factor varies and others remain fixed. In the natural sciences, they help express sensitivity in models with many interacting variables. They are also central to statistical and data-driven methods that analyze multivariate relationships.
6.4 Partial differential equations
Partial differential equations, or PDEs, are equations involving unknown functions and their partial derivatives. They model a wide range of phenomena, including diffusion, waves, and steady-state processes. The study of PDEs relies heavily on the properties and interpretation of partial derivatives.
7 Advanced topics
More advanced theory addresses when partial derivatives exist, how they relate to continuity, and how they behave in broader geometric settings. These questions are important in rigorous analysis and differential geometry. They also clarify the limits of intuitive formulas.
7.1 Existence and continuity
A partial derivative may fail to exist at some points, even when the function is defined nearby. Continuity of the function does not guarantee differentiability, and differentiability is stronger than the mere existence of first partial derivatives. Careful conditions are often needed in proofs and applications.
7.2 Equality of mixed partials
The equality of mixed partials depends on regularity assumptions. When the relevant second partial derivatives are continuous in a neighborhood, the mixed derivatives usually agree. Without such conditions, the order of differentiation may matter.
7.3 Differentiability and smoothness
A function is differentiable in the multivariable sense when it can be well approximated by a linear map near a point. Smoothness refers to the existence of derivatives of many orders, often with continuity. These properties are stronger than simply having partial derivatives and are important for advanced calculus.
7.4 Partial derivatives on manifolds
On manifolds, partial derivatives are defined using local coordinates rather than global Cartesian variables. They describe change relative to a chosen coordinate chart and must be interpreted carefully across overlapping charts. This coordinate-based approach extends multivariable calculus to curved spaces.