Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

The partial derivative of a function \(f(x, y, \dots)\) with respect to the variable \(x\) (analogously for any other variable) is variously denoted by

It is the rate of change of the function in the \(x\)-direction.

Sometimes, for \(z=f(x, y, \ldots)\), the partial derivative of \(z\) with respect to \(x\) is denoted as \(\tfrac{\partial z}{\partial x}.\) Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:

\[f'_x(x, y, \ldots), \frac{\partial f}{\partial x} (x, y, \ldots).\]

The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.

Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of \(\R^n\) and \(f:U\to\R\) a function. The partial derivative of f at the point \(\mathbf{a}=(a_1, \ldots, a_n) \in U\) with respect to the i-th variable x_i is defined as

\[\begin{aligned} \frac{\partial }{\partial x_i }f(\mathbf{a}) & = \lim_{h \to 0} \frac{f(a_1, \ldots , a_{i-1}, a_i+h, a_{i+1}\, \ldots ,a_{n})\ - f(a_1, \ldots, a_i, \dots ,a_n)}{h} \\ & = \lim_{h \to 0} \frac{f(\mathbf{a}+h\mathbf{e}_i) - f(\mathbf{a})}{h} \end{aligned}\]

where \(\mathbf{e_i}\) is the unit vector of i-th variable x_i. In fact, the last equality shows that the partial derivative is just the directional derivative where the direction is the \(i\)-th standard basis vector.

Even if all partial derivatives \(\partial f / \partial x_i(a)\) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C function. This can be used to generalize for vector valued functions, \(f:U \to \R^m\), by carefully using a component-wise argument.

The partial derivative \(\frac{\partial f}{\partial x}\) is itself a function defined on U and can be partially-differentiated again. If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:

\[\frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.\]

For the following examples, let f be a function in x, y, and z.

First-order partial derivatives:

\[\frac{ \partial f}{ \partial x} = f'_x = \partial_x f.\]

Second-order partial derivatives:

\[\frac{ \partial^2 f}{ \partial x^2} = f''_{xx} = \partial_{xx} f = \partial_x^2 f.\]

Second-order mixed derivatives:

\[\frac{\partial^2 f}{\partial y \,\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = (f'_{x})'_{y} = f''_{xy} = \partial_{yx} f = \partial_y \partial_x f .\]

Condensed: the full section is in Wikipedia.

An important example of a function of several variables is the case of a scalar-valued function \(f(x_1, \ldots, x_n)\) on a domain in Euclidean space \(\R^n\) (e.g., on \(\R^2\) or \(\R^3\)). In this case f has a partial derivative \(\partial f/\partial x_j\) with respect to each variable x_j. At the point a, these partial derivatives define the vector

\[\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).\]

This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently, the gradient produces a vector field.

A common abuse of notation is to define the del operator (∇) as follows in three-dimensional Euclidean space \(\R^3\) with unit vectors \(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\):

\[\nabla = \left[{\frac{\partial}{\partial x}} \right] \hat{\mathbf{i}} + \left[{\frac{\partial}{\partial y}} \right] \hat{\mathbf{j}} + \left[{\frac{\partial}{\partial z}}\right] \hat{\mathbf{k}}\]

Or, more generally, for n-dimensional Euclidean space \(\R^n\) with coordinates \(x_1, \ldots, x_n\) and unit vectors \(\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n\):

\[\nabla = \sum_{j=1}^n \left[\frac{\partial}{\partial x_j} \right] \hat{\mathbf{e}}_j = \left[\frac{\partial}{\partial x_1} \right] \hat{\mathbf{e}}_1 + \left[\frac{\partial}{\partial x_2} \right] \hat{\mathbf{e}}_2 + \dots + \left[\frac{\partial}{\partial x_n} \right] \hat{\mathbf{e}}_n\]

The directional derivative of a scalar function \[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n)\] along a vector \[\mathbf{v} = (v_1, \ldots, v_n)\] is the function \(\nabla_{\mathbf{v}}{f}\) defined by the limit \[\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}} = \left. \frac{\mathrm{d}}{\mathrm{d}t}f(\mathbf{x}+t\mathbf{v})\right|_{t=0}.\]

This definition is valid in a broad range of contexts, for example, where the norm of a vector is defined. In finite dimensions, it does not depend on the choice of norm, since all norms are equivalent. Its applicability extends to functions on finite-dimensional vector spaces without a metric and to differentiable manifolds, such as in general relativity.

Suppose that f is a function of more than one variable. For instance,

\[z = f(x,y) = x^2 + xy + y^2 .\]

The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the xz-plane, and those that are parallel to the yz-plane (which result from holding either y or x constant, respectively).

To find the slope of the line tangent to the function at P(1, 1) and parallel to the xz-plane, we treat y as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane y = 1. By finding the derivative of the equation while assuming that y is a constant, we find that the slope of f at the point (x, y) is:

\[\frac{\partial z}{\partial x} = 2x+y.\]

So at (1, 1), by substitution, the slope is 3. Therefore,

\[\frac{\partial z}{\partial x} = 3\]

Condensed: the full section is in Wikipedia.

Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function \(f(x, y, ...)\) the "own" second partial derivative with respect to x is simply the partial derivative of the partial derivative (both with respect to x):

\[\frac{\partial ^2 f}{\partial x^2} \equiv \partial \frac{{\partial f / \partial x}}{{\partial x}} \equiv \frac{{\partial f_x }}{{\partial x }} \equiv f_{xx}.\]

The cross partial derivative with respect to x and y is obtained by taking the partial derivative of f with respect to x, and then taking the partial derivative of the result with respect to y, to obtain

\[\frac{\partial ^2 f}{\partial y\, \partial x} \equiv \partial \frac{\partial f / \partial x}{\partial y} \equiv \frac{\partial f_x}{\partial y} \equiv f_{xy}.\]

Schwarz's theorem states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is,

\[\frac {\partial ^2 f}{\partial x\, \partial y} = \frac{\partial ^2 f}{\partial y\, \partial x}\]

or equivalently \(f_{yx} = f_{xy}.\)

Condensed: the full section is in Wikipedia.

There is a concept for partial derivatives that is analogous to antiderivatives for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of

\[\frac{\partial z}{\partial x} = 2x+y.\]

The so-called partial integral can be taken with respect to x (treating y as constant, in a similar manner to partial differentiation):

\[z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y).\]

Here, the constant of integration is no longer a constant, but instead a function of all the variables of the original function except x. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve x will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the constant represent an unknown function of all the other variables.

Thus the set of functions \(x^2 + xy + g(y)\), where g is any one-argument function, represents the entire set of functions in variables x, y that could have produced the x-partial derivative \(2x + y\).

Condensed: the full section is in Wikipedia.

The volume V of a cone depends on the cone's height h and its radius r according to the formula

\[V(r, h) = \frac{\pi r^2 h}{3}.\]

The partial derivative of V with respect to r is

\[\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},\]

which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h equals \(\frac{1}{3}\pi r^2\), which represents the rate with which the volume changes if its height is varied and its radius is kept constant.

By contrast, the total derivative of V with respect to r and h are respectively

\[\begin{aligned} \frac{dV}{dr} &= \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{dh}{dr}\,, \\ \frac{dV}{dh} &= \overbrace{\frac{\pi r^2}{3}}^\frac{\partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{dr}{dh}\,. \end{aligned}\]

Condensed: the full section is in Wikipedia.

Partial derivatives appear in any calculus-based optimization problem with more than one choice variable. For example, in economics a firm may wish to maximize profit π(x, y) with respect to the choice of the quantities x and y of two different types of output. The first order conditions for this optimization are π_x = 0 = π_y. Since both partial derivatives π_x and π_y will generally themselves be functions of both arguments x and y, these two first order conditions form a system of two equations in two unknowns.

Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as in Schrödinger wave equation, as well as in other equations from mathematical physics. The variables being held constant in partial derivatives here can be ratios of simple variables like mole fractions x_i in the following example involving the Gibbs energies in a ternary mixture system:

\[\bar{G_2}= G + (1-x_2) \left(\frac{{\partial G}}{{\partial x_2}}\right)_{\frac{x_1}{x_3}}\]

Express mole fractions of a component as functions of other components' mole fraction and binary mole ratios:

\(\begin{aligned} x_1 &= \frac{1-x_2}{1+\frac{x_3}{x_1}} \\ x_3 &= \frac{1-x_2}{1+\frac{x_1}{x_3}} \end{aligned}\)

Differential quotients can be formed at constant ratios like those above:

\[\begin{aligned} \left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_1}{1-x_2} \\ \left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_3}{1-x_2} \end{aligned}\]

Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:

Condensed: the full section is in Wikipedia.

Partial derivatives are key to target-aware image resizing algorithms. Widely known as seam carving, these algorithms require each pixel in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The algorithm then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of gradient at a pixel) depends heavily on the constructs of partial derivatives.

Partial derivatives play a prominent role in economics, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal consumption function may describe the amount spent on consumer goods as depending on both income and wealth; the marginal propensity to consume is then the partial derivative of the consumption function with respect to income.