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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x, y, \dots)} with respect to the variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} (analogously for any other variable) is variously denoted by
It is the rate of change of the function in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -direction.
Sometimes, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=f(x, y, \ldots)} , the partial derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is denoted as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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,[1] 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.[2]
Definition
[edit | edit source]Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:U\to\R} a function. The partial derivative of f at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}=(a_1, \ldots, a_n) \in U} with respect to the i-th variable xi is defined as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{e_i}} is the unit vector of i-th variable xi. In fact, the last equality shows that the partial derivative is just the directional derivative where the direction is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} -th standard basis vector.
Even if all partial derivatives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 C1 function. This can be used to generalize for vector valued functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:U \to \R^m} , by carefully using a component-wise argument.
The partial derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \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 C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.}
Notation
[edit | edit source]For the following examples, let f be a function in x, y, and z.
First-order partial derivatives:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial f}{ \partial x} = f'_x = \partial_x f.}
Second-order partial derivatives:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ \partial^2 f}{ \partial x^2} = f''_{xx} = \partial_{xx} f = \partial_x^2 f.}
Second-order mixed derivatives:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 .}
Higher-order partial and mixed derivatives:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^{i+j+k} f}{\partial x^i \partial y^j \partial z^k } = f^{(i, j, k)} = \partial_x^i \partial_y^j \partial_z^k f.}
When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{\partial f}{\partial x} \right)_{y,z} .}
Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of the function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f(x,y,z)}{\partial x}}
is used for the function, while
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f(u,v,w)}{\partial u}}
might be used for the value of the function at the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)=(u,v,w)} . However, this convention breaks down when we want to evaluate the partial derivative at a point like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)=(17, u+v, v^2)} . In such a case, evaluation of the function must be expressed in an unwieldy manner as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f(x,y,z)}{\partial x}(17, u+v, v^2)}
or
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \frac{\partial f(x,y,z)}{\partial x}\right |_{(x,y,z)=(17, u+v, v^2)}}
in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_i} as the partial derivative symbol with respect to the i-th variable. For instance, one would write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1 f(17, u+v, v^2)} for the example described above, while the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1 f} represents the partial derivative function with respect to the first variable.[3]
For higher order partial derivatives, the partial derivative (function) of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_i f} with respect to the j-th variable is denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_j(D_i f)=D_{i,j} f} . That is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_j\circ D_i =D_{i,j}} , so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied.
Gradient
[edit | edit source]An important example of a function of several variables is the case of a scalar-valued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_1, \ldots, x_n)} on a domain in Euclidean space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} (e.g., on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^3} ). In this case f has a partial derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial f/\partial x_j} with respect to each variable xj. At the point a, these partial derivatives define the vector
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^3} with unit vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}} :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} with coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1, \ldots, x_n} and unit vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n} :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}
Directional derivative
[edit | edit source]Example
[edit | edit source]Suppose that f is a function of more than one variable. For instance,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x} = 2x+y.}
So at (1, 1), by substitution, the slope is 3. Therefore,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x} = 3}
at the point (1, 1). That is, the partial derivative of z with respect to x at (1, 1) is 3, as shown in the graph.
The function f can be reinterpreted as a family of functions of one variable indexed by the other variables:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y) = f_y(x) = x^2 + xy + y^2.}
In other words, every value of y defines a function, denoted fy, which is a function of one variable x.[4] That is,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_y(x) = x^2 + xy + y^2.}
In this section the subscript notation fy denotes a function contingent on a fixed value of y, and not a partial derivative.
Once a value of y is chosen, say a, then f(x,y) determines a function fa which traces a curve x2 + ax + a2 on the xz-plane:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_a(x) = x^2 + ax + a^2.}
In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that being x. Consequently, the definition of the derivative for a function of one variable applies:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_a'(x) = 2x + a.}
The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function which describes the variation of f in the x direction:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial x}(x,y) = 2x + y.}
This is the partial derivative of f with respect to x. Here '∂' is a rounded 'd' called the partial derivative symbol; to distinguish it from the letter 'd', '∂' is sometimes pronounced "partial".
Higher order partial derivatives
[edit | edit source]Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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):[5]: 316–318
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {\partial ^2 f}{\partial x\, \partial y} = \frac{\partial ^2 f}{\partial y\, \partial x}}
or equivalently Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{yx} = f_{xy}.}
Own and cross partial derivatives appear in the Hessian matrix which is used in the second order conditions in optimization problems. The higher order partial derivatives can be obtained by successive differentiation
Antiderivative analogue
[edit | edit source]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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x + y} .
If all the partial derivatives of a function are known (for example, with the gradient), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is conservative.
Applications
[edit | edit source]Geometry
[edit | edit source]
The volume V of a cone depends on the cone's height h and its radius r according to the formula
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r, h) = \frac{\pi r^2 h}{3}.}
The partial derivative of V with respect to r is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}
The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.
If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \frac{h}{r} = \frac{dh}{dr}.}
This gives the total derivative with respect to r,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dV}{dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k\,,}
which simplifies to
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dV}{dr} = k \pi r^2,}
Similarly, the total derivative with respect to h is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dV}{dh} = \pi r^2.}
The total derivative with respect to both r and h of the volume intended as scalar function of these two variables is given by the gradient vector
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla V = \left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}\right) = \left(\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2\right).}
Optimization
[edit | edit source]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 πx = 0 = πy. Since both partial derivatives πx and πy 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.
Thermodynamics, quantum mechanics and mathematical physics
[edit | edit source]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 xi in the following example involving the Gibbs energies in a ternary mixture system:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \begin{align} 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{align}}
Differential quotients can be formed at constant ratios like those above:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}
Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} X &= \frac{x_3}{x_1 + x_3} \\ Y &= \frac{x_3}{x_2 + x_3} \\ Z &= \frac{x_2}{x_1 + x_2} \end{align}}
which can be used for solving partial differential equations like:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{\partial \mu_2}{\partial n_1}\right)_{n_2, n_3} = \left(\frac{\partial \mu_1}{\partial n_2}\right)_{n_1, n_3}}
This equality can be rearranged to have differential quotient of mole fractions on one side.
Image resizing
[edit | edit source]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.
Economics
[edit | edit source]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.
See also
[edit | edit source]- d'Alembert operator
- Chain rule
- Curl (mathematics)
- Divergence
- Exterior derivative
- Iterated integral
- Jacobian matrix and determinant
- Laplace operator
- Multivariable calculus
- Notation for differentiation
- Partial differential
- Symmetry of second derivatives
- Triple product rule, also known as the cyclic chain rule.
Notes
[edit | edit source]- ↑ Cajori, Florian (1952), A History of Mathematical Notations, 2 (3 ed.), The Open Court Publishing Company, 596
- ↑ Miller, Jeff (n.d.). "Earliest Uses of Symbols of Calculus". In O'Connor, John J.; Robertson, Edmund F. (eds.). MacTutor History of Mathematics archive. University of St Andrews. Retrieved 2023-06-15.
- ↑ Spivak, M. (1965). Calculus on Manifolds. New York: W. A. Benjamin. p. 44. ISBN 9780805390216.
- ↑ This can also be expressed as the adjointness between the product space and function space constructions.
- ↑ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill.