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Partial derivative

From Wikipedia, the free encyclopedia
Derivative of a function with multiple variables
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Inmathematics, apartial derivative of afunction of several variables is itsderivative with respect to one of those variables, with the others held constant (as opposed to thetotal derivative, in which all variables are allowed to vary). Partial derivatives are used invector calculus anddifferential geometry.

The partial derivative of a functionf(x,y,…){\displaystyle f(x,y,\dots )} with respect to the variablex{\displaystyle x} is variously denoted by

fx{\displaystyle f_{x}},fx′{\displaystyle f'_{x}},∂xf{\displaystyle \partial _{x}f}, Dxf{\displaystyle \ D_{x}f},D1f{\displaystyle D_{1}f},∂∂xf{\displaystyle {\frac {\partial }{\partial x}}f}, or∂f∂x{\displaystyle {\frac {\partial f}{\partial x}}}.

It can be thought of as the rate of change of the function in thex{\displaystyle x}-direction.

Sometimes, forz=f(x,y,…){\displaystyle z=f(x,y,\ldots )}, the partial derivative ofz{\displaystyle z} with respect tox{\displaystyle x} is denoted as∂z∂x.{\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:

fx′(x,y,…),∂f∂x(x,y,…).{\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 byMarquis de Condorcet from 1770,[1] who used it forpartial differences. The modern partial derivative notation was created byAdrien-Marie Legendre (1786), although he later abandoned it;Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.[2]

Definition

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Like ordinary derivatives, the partial derivative is defined as alimit. LetU be anopen subset ofRn{\displaystyle \mathbb {R} ^{n}} andf:U→R{\displaystyle f:U\to \mathbb {R} } a function. The partial derivative off at the pointa=(a1,…,an)∈U{\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to thei-th variablexi is defined as

∂∂xif(a)=limh→0f(a1,…,ai−1,ai+h,ai+1…,an) −f(a1,…,ai,…,an)h=limh→0f(a+hei)−f(a)h.{\displaystyle {\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}}}

Whereei{\displaystyle \mathbf {e_{i}} } is theunit vector ofi-th variablexi. Even if all partial derivatives∂f/∂xi(a){\displaystyle \partial f/\partial x_{i}(a)} exist at a given pointa, the function need not becontinuous there. However, if all partial derivatives exist in aneighborhood ofa and are continuous there, thenf istotally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said thatf is aC1 function. This can be used to generalize for vector valued functions,f:U→Rm{\displaystyle f:U\to \mathbb {R} ^{m}}, by carefully using a componentwise argument.

The partial derivative∂f∂x{\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined onU and can again be partially differentiated. If the direction of derivative isnot repeated, it is called amixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set),f is termed aC2 function at that point (or on that set); in this case, the partial derivatives can be exchanged byClairaut's theorem:

∂2f∂xi∂xj=∂2f∂xj∂xi.{\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.}

Notation

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Further information:∂

For the following examples, letf be a function inx,y, andz.

First-order partial derivatives:

∂f∂x=fx′=∂xf.{\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.}

Second-order partial derivatives:

∂2f∂x2=fxx″=∂xxf=∂x2f.{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f.}

Second-ordermixed derivatives:

∂2f∂y∂x=∂∂y(∂f∂x)=(fx′)y′=fxy″=∂yxf=∂y∂xf.{\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:

∂i+j+kf∂xi∂yj∂zk=f(i,j,k)=∂xi∂yj∂zkf.{\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 asstatistical mechanics, the partial derivative off with respect tox, holdingy andz constant, is often expressed as

(∂f∂x)y,z.{\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}.}

Conventionally, for clarity and simplicity of notation, the partial derivativefunction and thevalue of the function at a specific point areconflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like

∂f(x,y,z)∂x{\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}}

is used for the function, while

∂f(u,v,w)∂u{\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}}

might be used for the value of the function at the point(x,y,z)=(u,v,w){\displaystyle (x,y,z)=(u,v,w)}. However, this convention breaks down when we want to evaluate the partial derivative at a point like(x,y,z)=(17,u+v,v2){\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

∂f(x,y,z)∂x(17,u+v,v2){\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})}

or

∂f(x,y,z)∂x|(x,y,z)=(17,u+v,v2){\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 withDi{\displaystyle D_{i}} as the partial derivative symbol with respect to thei-th variable. For instance, one would writeD1f(17,u+v,v2){\displaystyle D_{1}f(17,u+v,v^{2})} for the example described above, while the expressionD1f{\displaystyle D_{1}f} represents the partial derivativefunction with respect to the first variable.[3]

For higher order partial derivatives, the partial derivative (function) ofDif{\displaystyle D_{i}f} with respect to thej-th variable is denotedDj(Dif)=Di,jf{\displaystyle D_{j}(D_{i}f)=D_{i,j}f}. That is,Dj∘Di=Di,j{\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 thatDi,j=Dj,i{\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions onf are satisfied.

Gradient

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Main article:Gradient

An important example of a function of several variables is the case of ascalar-valued functionf(x1,…,xn){\displaystyle f(x_{1},\ldots ,x_{n})} on a domain in Euclidean spaceRn{\displaystyle \mathbb {R} ^{n}} (e.g., onR2{\displaystyle \mathbb {R} ^{2}} orR3{\displaystyle \mathbb {R} ^{3}}). In this casef has a partial derivative∂f/∂xj{\displaystyle \partial f/\partial x_{j}} with respect to each variablexj. At the pointa, these partial derivatives define the vector

∇f(a)=(∂f∂x1(a),…,∂f∂xn(a)).{\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 thegradient off ata. Iff is differentiable at every point in some domain, then the gradient is a vector-valued function∇f which takes the pointa to the vector∇f(a). Consequently, the gradient produces avector field.

A commonabuse of notation is to define thedel operator (∇) as follows in three-dimensionalEuclidean spaceR3{\displaystyle \mathbb {R} ^{3}} withunit vectorsi^,j^,k^{\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}}:

∇=[∂∂x]i^+[∂∂y]j^+[∂∂z]k^{\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, forn-dimensional Euclidean spaceRn{\displaystyle \mathbb {R} ^{n}} with coordinatesx1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} and unit vectorse^1,…,e^n{\displaystyle {\hat {\mathbf {e} }}_{1},\ldots ,{\hat {\mathbf {e} }}_{n}}:

∇=∑j=1n[∂∂xj]e^j=[∂∂x1]e^1+[∂∂x2]e^2+⋯+[∂∂xn]e^n{\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

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This section is an excerpt fromDirectional derivative § Definition.[edit]
Acontour plot off(x,y)=x2+y2{\displaystyle f(x,y)=x^{2}+y^{2}}, showing the gradient vector in black, and the unit vectoru{\displaystyle \mathbf {u} } scaled by the directional derivative in the direction ofu{\displaystyle \mathbf {u} } in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.

Thedirectional derivative of ascalar functionf(x)=f(x1,x2,…,xn){\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})}along a vectorv=(v1,…,vn){\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}is thefunction∇vf{\displaystyle \nabla _{\mathbf {v} }{f}} defined by thelimit[4]∇vf(x)=limh→0f(x+hv)−f(x)h||v||=1||v||ddtf(x+tv)|t=0.{\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h||\mathbf {v} ||}}=\left.{\frac {1}{||\mathbf {v} ||}}{\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 thenorm of a vector (and hence a unit vector) is defined.[5]

Example

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

z=f(x,y)=x2+xy+y2.{\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.}

A graph ofz =x2 +xy +y2. For the partial derivative at(1, 1) that leavesy constant, the correspondingtangent line is parallel to thexz-plane.
A slice of the graph above showing the function in thexz-plane aty = 1. The two axes are shown here with different scales. The slope of the tangent line is 3.

Thegraph of this function defines asurface inEuclidean space. To every point on this surface, there are an infinite number oftangent lines. Partial differentiation is the act of choosing one of these lines and finding itsslope. Usually, the lines of most interest are those that are parallel to thexz-plane, and those that are parallel to theyz-plane (which result from holding eithery orx constant, respectively).

To find the slope of the line tangent to the function atP(1, 1) and parallel to thexz-plane, we treaty as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the planey = 1. By finding thederivative of the equation while assuming thaty is a constant, we find that the slope off at the point(x,y) is:

∂z∂x=2x+y.{\displaystyle {\frac {\partial z}{\partial x}}=2x+y.}

So at(1, 1), by substitution, the slope is3. Therefore,

∂z∂x=3{\displaystyle {\frac {\partial z}{\partial x}}=3}

at the point(1, 1). That is, the partial derivative ofz with respect tox at(1, 1) is3, as shown in the graph.

The functionf can be reinterpreted as a family of functions of one variable indexed by the other variables:

f(x,y)=fy(x)=x2+xy+y2.{\displaystyle f(x,y)=f_{y}(x)=x^{2}+xy+y^{2}.}

In other words, every value ofy defines a function, denotedfy, which is a function of one variablex.[6] That is,

fy(x)=x2+xy+y2.{\displaystyle f_{y}(x)=x^{2}+xy+y^{2}.}

In this section the subscript notationfy denotes a function contingent on a fixed value ofy, and not a partial derivative.

Once a value ofy is chosen, saya, thenf(x,y) determines a functionfa which traces a curvex2 +ax +a2 on thexz-plane:

fa(x)=x2+ax+a2.{\displaystyle f_{a}(x)=x^{2}+ax+a^{2}.}

In this expression,a is aconstant, not avariable, sofa is a function of only one real variable, that beingx. Consequently, the definition of the derivative for a function of one variable applies:

fa′(x)=2x+a.{\displaystyle f_{a}'(x)=2x+a.}

The above procedure can be performed for any choice ofa. Assembling the derivatives together into a function gives a function which describes the variation off in thex direction:

∂f∂x(x,y)=2x+y.{\displaystyle {\frac {\partial f}{\partial x}}(x,y)=2x+y.}

This is the partial derivative off with respect tox. Here '∂' is a rounded 'd' called thepartial derivative symbol; to distinguish it from the letter 'd', '∂' is sometimes pronounced "partial".

Higher order partial derivatives

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Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the functionf(x,y,...){\displaystyle f(x,y,...)} the "own" second partial derivative with respect tox is simply the partial derivative of the partial derivative (both with respect tox):[7]: 316–318 

∂2f∂x2≡∂∂f/∂x∂x≡∂fx∂x≡fxx.{\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 tox andy is obtained by taking the partial derivative off with respect tox, and then taking the partial derivative of the result with respect toy, to obtain

∂2f∂y∂x≡∂∂f/∂x∂y≡∂fx∂y≡fxy.{\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,

∂2f∂x∂y=∂2f∂y∂x{\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}}={\frac {\partial ^{2}f}{\partial y\,\partial x}}}

or equivalentlyfyx=fxy.{\displaystyle f_{yx}=f_{xy}.}

Own and cross partial derivatives appear in theHessian matrix which is used in thesecond order conditions inoptimization problems.The higher order partial derivatives can be obtained by successive differentiation

Antiderivative analogue

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

Consider the example of

∂z∂x=2x+y.{\displaystyle {\frac {\partial z}{\partial x}}=2x+y.}

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

z=∫∂z∂xdx=x2+xy+g(y).{\displaystyle z=\int {\frac {\partial z}{\partial x}}\,dx=x^{2}+xy+g(y).}

Here, theconstant of integration is no longer a constant, but instead a function of all the variables of the original function exceptx. 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 involvex 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 functionsx2+xy+g(y){\displaystyle x^{2}+xy+g(y)}, whereg is any one-argument function, represents the entire set of functions in variablesx,y that could have produced thex-partial derivative2x+y{\displaystyle 2x+y}.

If all the partial derivatives of a function are known (for example, with thegradient), 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 isconservative.

Applications

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Geometry

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The volume of a cone depends on height and radius

ThevolumeV of acone depends on the cone'sheighth and itsradiusr according to the formula

V(r,h)=πr2h3.{\displaystyle V(r,h)={\frac {\pi r^{2}h}{3}}.}

The partial derivative ofV with respect tor is

∂V∂r=2πrh3,{\displaystyle {\frac {\partial V}{\partial r}}={\frac {2\pi rh}{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 toh equals13πr2{\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, thetotal derivative ofV with respect tor andh are respectively

dVdr=2πrh3⏞∂V∂r+πr23⏞∂V∂hdhdr,dVdh=πr23⏞∂V∂h+2πrh3⏞∂V∂rdrdh.{\displaystyle {\begin{aligned}{\frac {dV}{dr}}&=\overbrace {\frac {2\pi rh}{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 rh}{3}} ^{\frac {\partial V}{\partial r}}{\frac {dr}{dh}}\,.\end{aligned}}}

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 ratiok,

k=hr=dhdr.{\displaystyle k={\frac {h}{r}}={\frac {dh}{dr}}.}

This gives the total derivative with respect tor,

dVdr=2πrh3+πr23k,{\displaystyle {\frac {dV}{dr}}={\frac {2\pi rh}{3}}+{\frac {\pi r^{2}}{3}}k\,,}

which simplifies to

dVdr=kπr2,{\displaystyle {\frac {dV}{dr}}=k\pi r^{2},}

Similarly, the total derivative with respect toh is

dVdh=πr2.{\displaystyle {\frac {dV}{dh}}=\pi r^{2}.}

The total derivative with respect tobothr andh of the volume intended as scalar function of these two variables is given by thegradient vector

∇V=(∂V∂r,∂V∂h)=(23πrh,13πr2).{\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

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Partial derivatives appear in any calculus-basedoptimization problem with more than one choice variable. For example, ineconomics a firm may wish to maximizeprofitπ(x,y) with respect to the choice of the quantitiesx andy of two different types of output. Thefirst order conditions for this optimization areπx = 0 = πy. Since both partial derivativesπx andπy will generally themselves be functions of both argumentsx andy, these two first order conditions form asystem of two equations in two unknowns.

Thermodynamics, quantum mechanics and mathematical physics

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Partial derivatives appear in thermodynamic equations likeGibbs-Duhem equation, in quantum mechanics as inSchrödinger wave equation, as well as in other equations frommathematical physics. The variables being held constant in partial derivatives here can be ratios of simple variables likemole fractionsxi in the following example involving the Gibbs energies in a ternary mixture system:

G2¯=G+(1−x2)(∂G∂x2)x1x3{\displaystyle {\bar {G_{2}}}=G+(1-x_{2})\left({\frac {\partial G}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}}

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

x1=1−x21+x3x1x3=1−x21+x1x3{\textstyle {\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:

(∂x1∂x2)x1x3=−x11−x2(∂x3∂x2)x1x3=−x31−x2{\displaystyle {\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:

X=x3x1+x3Y=x3x2+x3Z=x2x1+x2{\displaystyle {\begin{aligned}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{aligned}}}

which can be used for solvingpartial differential equations like:

(∂μ2∂n1)n2,n3=(∂μ1∂n2)n1,n3{\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

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

Economics

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

See also

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Notes

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  1. ^Cajori, Florian (1952),A History of Mathematical Notations, vol. 2 (3 ed.), The Open Court Publishing Company, 596
  2. ^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. Retrieved2023-06-15.
  3. ^Spivak, M. (1965).Calculus on Manifolds. New York: W. A. Benjamin. p. 44.ISBN 9780805390216.
  4. ^R. Wrede; M.R. Spiegel (2010).Advanced Calculus (3rd ed.). Schaum's Outline Series.ISBN 978-0-07-162366-7.
  5. ^The applicability extends to functions over spaces without ametric and todifferentiable manifolds, such as ingeneral relativity.
  6. ^This can also be expressed as theadjointness between theproduct space andfunction space constructions.
  7. ^Chiang, Alpha C. (1984).Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill.

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