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Differential of a function

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Notion in calculus

For other uses of "differential" in mathematics, seeDifferential (mathematics).

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\int _{a}^{b}f'(t)\,dt=f(b)-f(a)

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Incalculus, thedifferential represents theprincipal part of the change in afunction $y=f(x)$ with respect to changes in the independent variable. The differential $d y {\displaystyle dy}$ is defined by $dy=f'(x)\,dx,$ where $f'(x)$ is thederivative off with respect to $x {\displaystyle x}$ , and $d x {\displaystyle dx}$ is an additional realvariable (so that $d y {\displaystyle dy}$ is a function of $x {\displaystyle x}$ and $d x {\displaystyle dx}$ ). The notation is such that the equation

$dy={\frac {dy}{dx}}\,dx$

holds, where the derivative is represented in theLeibniz notation $dy/dx$ , and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

$df(x)=f'(x)\,dx.$

The precise meaning of the variables $d y {\displaystyle dy}$ and $d x {\displaystyle dx}$ depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particulardifferential form, or analytical significance if the differential is regarded as alinear approximation to the increment of a function. Traditionally, the variables $d x {\displaystyle dx}$ and $d y {\displaystyle dy}$ are considered to be very small (infinitesimal), and this interpretation is made rigorous innon-standard analysis.

History and usage

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The differential was first introduced via an intuitive or heuristic definition byIsaac Newton and furthered byGottfried Leibniz, who thought of the differential dy as an infinitely small (orinfinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x. For that reason, the instantaneous rate of change ofy with respect tox, which is the value of thederivative of the function, is denoted by the fraction

${\frac {dy}{dx}}$ in what is called theLeibniz notation for derivatives. The quotient $dy/dx$ is not infinitely small; rather it is areal number.

The use of infinitesimals in this form was widely criticized, for instance by the famous pamphletThe Analyst by Bishop Berkeley.Augustin-Louis Cauchy (1823) defined the differential without appeal to the atomism of Leibniz's infinitesimals.^[1]^[2] Instead, Cauchy, followingd'Alembert, inverted the logical order of Leibniz and his successors: the derivative itself became the fundamental object, defined as alimit of difference quotients, and the differentials were then defined in terms of it. That is, one was free todefine the differential $d y {\displaystyle dy}$ by an expression $dy=f'(x)\,dx$ in which $d y {\displaystyle dy}$ and $d x {\displaystyle dx}$ are simply new variables taking finite real values,^[3] not fixed infinitesimals as they had been for Leibniz.^[4]

According toBoyer (1959, p. 12), Cauchy's approach was a significant logical improvement over the infinitesimal approach of Leibniz because, instead of invoking the metaphysical notion of infinitesimals, the quantities $d y {\displaystyle dy}$ and $d x {\displaystyle dx}$ could now be manipulated in exactly the same manner as any other real quantitiesin a meaningful way. Cauchy's overall conceptual approach to differentials remains the standard one in modern analytical treatments,^[5] although the final word on rigor, a fully modern notion of the limit, was ultimately due toKarl Weierstrass.^[6]

In physical treatments, such as those applied to the theory ofthermodynamics, the infinitesimal view still prevails.Courant & John (1999, p. 184) reconcile the physical use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the particular purpose for which they are intended. Thus "physical infinitesimals" need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense.

Following twentieth-century developments inmathematical analysis anddifferential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. Inreal analysis, it is more desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is alinear functional of an increment $\Delta x$ . This approach allows the differential (as a linear map) to be developed for a variety of more sophisticated spaces, ultimately giving rise to such notions as theFréchet orGateaux derivative. Likewise, indifferential geometry, the differential of a function at a point is a linear function of atangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: theexterior derivative of the function. Innon-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing (seedifferential (infinitesimal)).

Definition

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The differential of a function $f(x)$ at a point $x_{0}$ .

{\displaystyle f(x)} — The differential of a function $f(x)$ at a point $x_{0}$ .

The differential is defined in modern treatments of differential calculus as follows.^[7] The differential of a function $f(x)$ of a single real variable $x {\displaystyle x}$ is the function $d f {\displaystyle df}$ of two independent real variables $x {\displaystyle x}$ and $\Delta x$ given by

$df(x,\Delta x)\ {\stackrel {\mathrm {def} }{=}}\ f'(x)\,\Delta x.$

One or both of the arguments may be suppressed, i.e., one may see $df(x)$ or simply $d f {\displaystyle df}$ . If $y=f(x)$ , the differential may also be written as $d y {\displaystyle dy}$ . Since $dx(x,\Delta x)=\Delta x$ , it is conventional to write $dx=\Delta x$ so that the following equality holds:

$df(x)=f'(x)\,dx$

This notion of differential is broadly applicable when alinear approximation to a function is sought, in which the value of the increment $\Delta x$ is small enough. More precisely, if $f {\displaystyle f}$ is adifferentiable function at $x {\displaystyle x}$ , then the difference in $y {\displaystyle y}$ -values

$\Delta y\ {\stackrel {\rm {def}}{=}}\ f(x+\Delta x)-f(x)$

satisfies

$\Delta y=f'(x)\,\Delta x+\varepsilon =df(x)+\varepsilon \,$

where the error $\varepsilon$ in the approximation satisfies $\varepsilon /\Delta x\rightarrow 0$ as $\Delta x\rightarrow 0$ . In other words, one has the approximate identity

$\Delta y\approx dy$

in which the error can be made as small as desired relative to $\Delta x$ by constraining $\Delta x$ to be sufficiently small; that is to say, ${\frac {\Delta y-dy}{\Delta x}}\to 0$ as $\Delta x\rightarrow 0$ . For this reason, the differential of a function is known as theprincipal (linear) part in the increment of a function: the differential is alinear function of the increment $\Delta x$ , and although the error $\varepsilon$ may be nonlinear, it tends to zero rapidly as $\Delta x$ tends to zero.

Differentials in several variables

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Operator / Function	$f(x)$	$f(x,y,u(x,y),v(x,y))$
Differential	1: $df\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}\,dx$	2: $d_{x}f\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}\,dx$ 3: $df\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}dx+f'_{y}dy+f'_{u}du+f'_{v}dv$
Partial derivative	$f'_{x}\,{\overset {\underset {\mathrm {(1)} }{}}{=}}\,{\frac {df}{dx}}$	$f'_{x}\,{\overset {\underset {\mathrm {(2)} }{}}{=}}\,{\frac {d_{x}f}{dx}}={\frac {\partial f}{\partial x}}$
Total derivative	${\frac {df}{dx}}\,{\overset {\underset {\mathrm {(1)} }{}}{=}}\,f'_{x}$	${\frac {df}{dx}}\,{\overset {\underset {\mathrm {(3)} }{}}{=}}\,f'_{x}+f'_{u}{\frac {du}{dx}}+f'_{v}{\frac {dv}{dx}};(f'_{y}{\frac {dy}{dx}}=0)$

FollowingGoursat (1904, I, §15), for functions of more than one independent variable,

$y=f(x_{1},\dots ,x_{n}),$

thepartial differential ofy with respect to any one of the variables x_i is the principal part of the change iny resulting from a change dx_i in that one variable. The partial differential is therefore

${\frac {\partial y}{\partial x_{i}}}dx_{i}$

involving thepartial derivative ofy with respect to x_i. The sum of the partial differentials with respect to all of the independent variables is thetotal differential

$dy={\frac {\partial y}{\partial x_{1}}}dx_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}dx_{n},$

which is the principal part of the change iny resulting from changes in the independent variables x_i.

More precisely, in the context of multivariable calculus, followingCourant (1937b), iff is a differentiable function, then by thedefinition of differentiability, the increment

${\begin{aligned}\Delta y&{}~{\stackrel {\mathrm {def} }{=}}~f(x_{1}+\Delta x_{1},\dots ,x_{n}+\Delta x_{n})-f(x_{1},\dots ,x_{n})\\&{}={\frac {\partial y}{\partial x_{1}}}\Delta x_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}\Delta x_{n}+\varepsilon _{1}\Delta x_{1}+\cdots +\varepsilon _{n}\Delta x_{n}\end{aligned}}$

where the error termsε_i tend to zero as the incrementsΔx_i jointly tend to zero. The total differential is then rigorously defined as

$dy={\frac {\partial y}{\partial x_{1}}}\Delta x_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}\Delta x_{n}.$

Since, with this definition, $dx_{i}(\Delta x_{1},\dots ,\Delta x_{n})=\Delta x_{i},$ one has $dy={\frac {\partial y}{\partial x_{1}}}\,dx_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}\,dx_{n}.$

As in the case of one variable, the approximate identity holds

$dy\approx \Delta y$

in which the total error can be made as small as desired relative to ${\textstyle {\sqrt {\Delta x_{1}^{2}+\cdots +\Delta x_{n}^{2}}}}$ by confining attention to sufficiently small increments.

Application of the total differential to error estimation

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In measurement, the total differential is used inestimating the error $\Delta f$ of a function $f {\displaystyle f}$ based on the errors $\Delta x,\Delta y,\ldots$ of the parameters $x,y,\ldots$ . Assuming that the interval is short enough for the change to be approximately linear:

$\Delta f(x)=f'(x)\Delta x$

and that all variables are independent, then for all variables,

$\Delta f=f_{x}\Delta x+f_{y}\Delta y+\cdots$

This is because the derivative $f_{x}$ with respect to the particular parameter $x {\displaystyle x}$ gives the sensitivity of the function $f {\displaystyle f}$ to a change in $x {\displaystyle x}$ , in particular the error $\Delta x$ . As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.:

Let

f(a,b)=ab

. Then, the finite error can be approximated as

$\Delta f=f_{a}\Delta a+f_{b}\Delta b.$ Evaluating the derivatives: $\Delta f=b\Delta a+a\Delta b.$ Dividing byf, which isa ×b

{\frac {\Delta f}{f}}={\frac {\Delta a}{a}}+{\frac {\Delta b}{b}}

That is to say, in multiplication, the totalrelative error is the sum of the relative errors of the parameters.

To illustrate how this depends on the function considered, consider the case where the function is $f(a,b)=a\ln b$ instead. Then, it can be computed that the error estimate is ${\frac {\Delta f}{f}}={\frac {\Delta a}{a}}+{\frac {\Delta b}{b\ln b}}$ with an extralnb factor not found in the case of a simple product. This additional factor tends to make the error smaller, as the denominatorb lnb is larger than a bare b.

Higher-order differentials

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Higher-order differentials of a functiony =f(x) of a single variablex can be defined via:^[8] $d^{2}y=d(dy)=d(f'(x)dx)=(df'(x))dx=f''(x)\,(dx)^{2},$ and, in general, $d^{n}y=f^{(n)}(x)\,(dx)^{n}.$ Informally, this motivates Leibniz's notation for higher-order derivatives $f^{(n)}(x)={\frac {d^{n}f}{dx^{n}}}.$ When the independent variablex itself is permitted to depend on other variables, then the expression becomes more complicated, as it must include also higher order differentials inx itself. Thus, for instance, ${\begin{aligned}d^{2}y&=f''(x)\,(dx)^{2}+f'(x)d^{2}x\\[1ex]d^{3}y&=f'''(x)\,(dx)^{3}+3f''(x)dx\,d^{2}x+f'(x)d^{3}x\end{aligned}}$ and so forth.

Similar considerations apply to defining higher order differentials of functions of several variables. For example, iff is a function of two variablesx andy, then $d^{n}f=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {\partial ^{n}f}{\partial x^{k}\partial y^{n-k}}}(dx)^{k}(dy)^{n-k},$ where ${\textstyle {\binom {n}{k}}}$ is abinomial coefficient. In more variables, an analogous expression holds, but with an appropriatemultinomial expansion rather than binomial expansion.^[9]

Higher order differentials in several variables also become more complicated when the independent variables are themselves allowed to depend on other variables. For instance, for a functionf ofx andy which are allowed to depend on auxiliary variables, one has $d^{2}f=\left({\frac {\partial ^{2}f}{\partial x^{2}}}(dx)^{2}+2{\frac {\partial ^{2}f}{\partial x\partial y}}dx\,dy+{\frac {\partial ^{2}f}{\partial y^{2}}}(dy)^{2}\right)+{\frac {\partial f}{\partial x}}d^{2}x+{\frac {\partial f}{\partial y}}d^{2}y.$

Because of this notational awkwardness, the use of higher order differentials was roundly criticized byHadamard (1935), who concluded:

Enfin, que signifie ou que représente l'égalité
$d^{2}z=r\,dx^{2}+2s\,dx\,dy+t\,dy^{2}\,?$
A mon avis, rien du tout.

That is:Finally, what is meant, or represented, by the equality [...]? In my opinion, nothing at all. In spite of this skepticism, higher order differentials did emerge as an important tool in analysis.^[10]

In these contexts, then-th order differential of the functionf applied to an incrementΔx is defined by $d^{n}f(x,\Delta x)=\left.{\frac {d^{n}}{dt^{n}}}f(x+t\Delta x)\right|_{t=0}$ or an equivalent expression, such as $\lim _{t\to 0}{\frac {\Delta _{t\Delta x}^{n}f}{t^{n}}}$ where $\Delta _{t\Delta x}^{n}f$ is annthforward difference with incrementtΔx.

This definition makes sense as well iff is a function of several variables (for simplicity taken here as a vector argument). Then then-th differential defined in this way is ahomogeneous function of degreen in the vector incrementΔx. Furthermore, theTaylor series off at the pointx is given by $f(x+\Delta x)\sim f(x)+df(x,\Delta x)+{\frac {1}{2}}d^{2}f(x,\Delta x)+\cdots +{\frac {1}{n!}}d^{n}f(x,\Delta x)+\cdots$ The higher orderGateaux derivative generalizes these considerations to infinite dimensional spaces.

Properties

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A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include:^[11]

Linearity: For constantsa andb and differentiable functionsf andg, $d(af+bg)=a\,df+b\,dg.$
Product rule: For two differentiable functionsf andg, $d(fg)=f\,dg+g\,df.$

An operationd with these two properties is known inabstract algebra as aderivation. They imply the power rule $d(f^{n})=nf^{n-1}df$ In addition, various forms of thechain rule hold, in increasing level of generality:^[12]

Ify =f(u) is a differentiable function of the variableu andu =g(x) is a differentiable function ofx, then $dy=f'(u)\,du=f'(g(x))g'(x)\,dx.$
Ify =f(x₁, ...,x_n) and all of the variables x₁, ...,x_n depend on another variable t, then by thechain rule for partial derivatives, one has ${\begin{aligned}dy={\frac {dy}{dt}}dt&={\frac {\partial y}{\partial x_{1}}}dx_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}dx_{n}\\[1ex]&={\frac {\partial y}{\partial x_{1}}}{\frac {dx_{1}}{dt}}\,dt+\cdots +{\frac {\partial y}{\partial x_{n}}}{\frac {dx_{n}}{dt}}\,dt.\end{aligned}}$ Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantitydt.
More general analogous expressions hold, in which the intermediate variablesx_i depend on more than one variable.

General formulation

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A consistent notion of differential can be developed for a functionf :Rⁿ →R^m between twoEuclidean spaces. Letx,Δx ∈Rⁿ be a pair ofEuclidean vectors. The increment in the functionf is $\Delta f=f(\mathbf {x} +\Delta \mathbf {x} )-f(\mathbf {x} ).$ If there exists anm ×nmatrixA such that $\Delta f=A\Delta \mathbf {x} +\|\Delta \mathbf {x} \|{\boldsymbol {\varepsilon }}$ in which the vectorε → 0 asΔx → 0, thenf is by definition differentiable at the pointx. The matrixA is sometimes known as theJacobian matrix, and thelinear transformation that associates to the incrementΔx ∈Rⁿ the vectorAΔx ∈R^m is, in this general setting, known as the differentialdf(x) off at the pointx. This is precisely theFréchet derivative, and the same construction can be made to work for a function between anyBanach spaces.

Another fruitful point of view is to define the differential directly as a kind ofdirectional derivative: $df(\mathbf {x} ,\mathbf {h} )=\lim _{t\to 0}{\frac {f(\mathbf {x} +t\mathbf {h} )-f(\mathbf {x} )}{t}}=\left.{\frac {d}{dt}}f(\mathbf {x} +t\mathbf {h} )\right|_{t=0},$ which is the approach already taken for defining higher order differentials (and is most nearly the definition set forth by Cauchy). Ift represents time andx position, thenh represents a velocity instead of a displacement as we have heretofore regarded it. This yields yet another refinement of the notion of differential: that it should be a linear function of a kinematic velocity. The set of all velocities through a given point of space is known as thetangent space, and sodf gives a linear function on the tangent space: adifferential form. With this interpretation, the differential off is known as theexterior derivative, and has broad application indifferential geometry because the notion of velocities and the tangent space makes sense on anydifferentiable manifold. If, in addition, the output value off also represents a position (in a Euclidean space), then a dimensional analysis confirms that the output value ofdf must be a velocity. If one treats the differential in this manner, then it is known as thepushforward since it "pushes" velocities from a source space into velocities in a target space.

Other approaches

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Main article:Differential (infinitesimal)

Although the notion of having an infinitesimal incrementdx is not well-defined in modernmathematical analysis, a variety of techniques exist for defining theinfinitesimal differential so that the differential of a function can be handled in a manner that does not clash with theLeibniz notation. These include:

Defining the differential as a kind ofdifferential form, specifically theexterior derivative of a function. The infinitesimal increments are then identified with vectors in thetangent space at a point. This approach is popular indifferential geometry and related fields, because it readily generalizes to mappings betweendifferentiable manifolds.
Differentials asnilpotent elements ofcommutative rings. This approach is popular inalgebraic geometry.^[13]
Differentials in smooth models of set theory. This approach is known assynthetic differential geometry orsmooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas fromtopos theory are used tohide the mechanisms by which nilpotent infinitesimals are introduced.^[14]
Differentials as infinitesimals inhyperreal number systems, which are extensions of the real numbers which contain invertible infinitesimals and infinitely large numbers. This is the approach ofnonstandard analysis pioneered byAbraham Robinson.^[15]

Examples and applications

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Differentials may be effectively used innumerical analysis to study the propagation of experimental errors in a calculation, and thus the overallnumerical stability of a problem (Courant 1937a). Suppose that the variablex represents the outcome of an experiment andy is the result of a numerical computation applied tox. The question is to what extent errors in the measurement ofx influence the outcome of the computation ofy. If thex is known to within Δx of its true value, thenTaylor's theorem gives the following estimate on the error Δy in the computation ofy: $\Delta y=f'(x)\Delta x+{\frac {(\Delta x)^{2}}{2}}f''(\xi )$ whereξ =x +θΔx for some0 <θ < 1. IfΔx is small, then the second order term is negligible, so that Δy is, for practical purposes, well-approximated bydy =f'(x) Δx.

The differential is often useful to rewrite adifferential equation ${\frac {dy}{dx}}=g(x)$ in the form $dy=g(x)\,dx,$ in particular when one wants toseparate the variables.

Notes

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^For a detailed historical account of the differential, seeBoyer 1959, especially page 275 for Cauchy's contribution on the subject. An abbreviated account appears inKline 1972, Chapter 40.
^Cauchy explicitly denied the possibility of actual infinitesimal and infinite quantities (Boyer 1959, pp. 273–275), and took the radically different point of view that "a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge to zero" (Cauchy 1823, p. 12; translation fromBoyer 1959, p. 273).
^Boyer 1959, p. 275
^Boyer 1959, p. 12: "The differentials as thus defined are only newvariables, and not fixed infinitesimals..."
^Courant 1937a, II, §9: "Here we remark merely in passing that it is possible to use this approximate representation of the increment $\Delta y$ by the linear expression $hf(x)$ to construct a logically satisfactory definition of a "differential", as was done by Cauchy in particular."
^Boyer 1959, p. 284
^See, for instance, the influential treatises ofCourant 1937a,Kline 1977,Goursat 1904, andHardy 1908. Tertiary sources for this definition include alsoTolstov 2001 andItô 1993, §106.
^Cauchy 1823. See also, for instance,Goursat 1904, I, §14.
^Goursat 1904, I, §14
^In particular toinfinite dimensional holomorphy (Hille & Phillips 1974) andnumerical analysis via the calculus offinite differences.
^Goursat 1904, I, §17
^Goursat 1904, I, §§14,16
^Eisenbud & Harris 1998.
^SeeKock 2006 andMoerdijk & Reyes 1991.
^SeeRobinson 1996 andKeisler 1986.