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Dalam kalkulus, diferensial menyatakan perubahan suatu fungsi ${\textstyle y=f(x)}$ terhadap perubahan variabel bebas. Diferensial ${\textstyle dy}$ didefinisikan dengan$$dy=f'(x)\,dx$$dengan $f'(x)$ menyatakan turunan $f$ terhadap $x$, sedangkan ${\textstyle dx}$ menyatakan suatu peubah real tambahan; sehingga ${\textstyle dy}$ merupakan suatu fungsi dari ${\textstyle x}$ dan ${\textstyle dx}$). Karena itu, notasi tersebut berlaku suatu persamaan$$dy={\frac {dy}{dx}}\,dx.$$Pada persamaan ini, turunan dinyatakan dengan notasi Leibniz ${\textstyle dy/dx}$, dan bentuk notasi turunan menyerupai hasil bagi diferensial. Alternatif dari notasi ini juga ditulis sebagai$$df(x)=f'(x)\,dx.$$

The precise meaning of the variables $dy$ and $dx$ 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 particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables $dx$ and $dy$ are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.

Sejarah dan penggunaan

Diferensial pertama kali diperkenalkan melalui definisi intuitif atau heuristik oleh Isaac Newton. Gottfried Leibniz kemudian melanjutkan pengembangan konsep tersebut lebih lanjut dengan memandang diferensial $dy$ sebagai perubahan dengan bilangan yang sangat kecil (infinitesimal) di nilai fungsi $y$, yang bersesuaian dengan perubahan infinitesimal $dx$ di argumen fungsi $x$. Oleh karena itu, tingkat perubahan sesaat (instantaneous rate of change) $y$ terhadap $x$, yang merupakan nilai dari turunan fungsi, ditulis menggunakan pecahan$${\frac {dy}{dx}}.$$Notasi turunan ini dikenal dengan sebutan notasi Leibniz. Pada notasi tersebut, hasil bagi $dy/dx$ bukanlah suatu infinitesimal, melainkan suatu bilangan real.

Penggunaan infinteismal pada notasi tersebut telah dikritik secara luas, seperti Bishop Berkeley dalam pamflet terkenalnya yang berjudul The Analyst. Augustin-Louis Cauchy (1823) mendefinisikan diferensial tanpa membandingkan atomism of Leibniz's infinitesimals. Instead, Cauchy, following d'Alembert, inverted the logical order of Leibniz and his successors: the derivative itself became the fundamental object, defined as a limit of difference quotients, and the differentials were then defined in terms of it. That is, one was free to define the differential $dy$ by an expression

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

in which $dy$ and $dx$ are simply new variables taking finite real values, not fixed infinitesimals as they had been for Leibniz.

According to (Boyer 1959, hlm. 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 $dy$ and $dx$ could now be manipulated in exactly the same manner as any other real quantities in a meaningful way. Cauchy's overall conceptual approach to differentials remains the standard one in modern analytical treatments, although the final word on rigor, a fully modern notion of the limit, was ultimately due to Karl Weierstrass.

In physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. (Courant & John 1999, hlm. 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 in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real 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 a linear 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 the Fréchet or Gateaux derivative. Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function. In non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing (see differential (infinitesimal)).

Definition

Diferensial fungsi $f(x)$ di suatu titik $x_{0}$.

Diferensial didefinisikan dalam perlakuan kalkulus diferensial yang modern. Definisi mengatakan bahwa diferensial fungsi $f(x)$ dari suatu peubah real $x$ merupakan fungsi $df$ dari dua peubah variabel bebas $x$ dan $\Delta x$. Ini ditulis sebagai$$df(x,\Delta x){\stackrel {\mathrm {def} }{=}}f'(x)\,\Delta x.$$

Pada definisi di atas, seseorang dapat memandangnya sebagai $df(x)$, atau sederhananya, $df$. Jika $y=f(x)$, maka diferensial dapat ditulis pula sebagai $dy$. Karena $dx(x,\Delta x)=\Delta x$, maka secara konvensional ditulis sebagai $dx=\Delta x$ sehingga berlaku persamaan berikut:$$df(x)=f'(x)\,dx.$$

Gagasan tersebut dapat diterima dengan luas, terutama saat mencari hampiran linear dari suatu fungsi dengan nilai dari pertambahan $\Delta x$ yang cukup kecil. Lebih tepatnya, jika $f$ adalah suatu fungsi yang terdiferensialkan di $x$, maka selisih dari nilai $y$$$\Delta y{\stackrel {\rm {def}}{=}}f(x+\Delta x)-f(x)$$memenuhi bahwa$$\Delta y=f'(x)\,\Delta x+\varepsilon =df(x)+\varepsilon \,.$$Catatan bahwa galat $\varepsilon$ pada hampiran tersebut memenuhi $\varepsilon /\Delta x\rightarrow 0$ ketika $\Delta x\rightarrow 0$. Dengan kata lain, identitas tersebut dihampiri sebagai $\Delta y\approx dy$ dengan galatnya dapat dibuat sekecil mungkin dengan $\Delta x$, yang dilakukan dengan cara memaksa nilai $\Delta x$ menjadi kecil, dalam artian,

${\frac {\Delta y-dy}{\Delta x}}\to 0$

ketika $\Delta x\rightarrow 0$. Oleh karena demikian, diferensial fungsi dikenal sebagai bagian (linear) utama dalam pertambahan suatu fungsi: diferensial adalah fungsi linear dari pertambahan $\Delta x$, dan walaupun galat $\varepsilon$ tak linear, galat tersebut cenderung drastis menuju ke nol ketika $\Delta x$ menuju ke nol.

Diferensial dalam beberapa variabel

Operator / Fungsi	$f(x)$	$f(x,y,u(x,y),v(x,y))$
Diferensial	1: $df\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}\,dx$	2: $d_{x}f\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}\,dx$ 3: $df\,{\overset {\underset {\mathrm {def} }{}}{=}}\,f'_{x}dx+f'_{y}dy+f'_{u}du+f'_{v}dv$
Turunan parsial	$f'_{x}\,{\overset {\underset {\mathrm {(1)} }{}}{=}}\,{\frac {df}{dx}}$	$f'_{x}\,{\overset {\underset {\mathrm {(2)} }{}}{=}}\,{\frac {d_{x}f}{dx}}={\frac {\partial f}{\partial x}}$
Turunan total	${\frac {df}{dx}}\,{\overset {\underset {\mathrm {(1)} }{}}{=}}\,f'_{x}$	${\frac {df}{dx}}\,{\overset {\underset {\mathrm {(3)} }{}}{=}}\,f'_{x}+f'_{u}{\frac {du}{dx}}+f'_{v}{\frac {dv}{dx}};(f'_{y}{\frac {dy}{dx}}=0)$

Mengutip dari (Goursat 1904, I, §15), untuk fungsi dari variabel yang berjumlah lebih dari satu, $$y=f(x_{1},\dots ,x_{n}),$$ turunan parsial dari $y$ terhadap salah satu variabel $x_{1}$ adalah bagian prinsip (principal part) dari perubahan di $y$, yang dihasilkan dari perubahan $dx_{1}$ di suatu variabel tersebut. Oleh karena itu, diferensial parsial $${\frac {\partial y}{\partial x_{1}}}dx_{1}$$ melibatkan turunan parsial dari $y$ terhadap  $x_{1}$. Jumlah dari diferensial parsial terhadap semua variabel bebas sama dengan diferensial total $$dy={\frac {\partial y}{\partial x_{1}}}dx_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}dx_{n},$$ yang merupakan bagian prinsip dari perubahan di $y$ yang dihasilkan dari perubahan di variabel bebas $x_{i}$.

Diferensial parsial dapat dijelaskan lebih rinci lagi. Dalam kalkulus multivariabel, (Courant 1937b) mengutip bahwa jika $f$ adalah fungsi terdiferensialkan, maka berdasarkan definisi keterdiferensialan, pertambahan $$\Delta y{\stackrel {\mathrm {def} }{=}}f(x_{1}+\Delta x_{1},\dots ,x_{n}+\Delta x_{n})-f(x_{1},\dots ,x_{n})={\frac {\partial y}{\partial x_{1}}}\Delta x_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}\Delta x_{n}+\varepsilon _{1}\Delta x_{1}+\cdots +\varepsilon _{n}\Delta x_{n},$$dengan suku galat $\varepsilon _{1}$ cenderung menuju ke nol saat pertambahan $\Delta x_{1}$ sama-sama cenderung menuju ke nol. Oleh karena itu, diferensial total didefinisikan secara cermat sebagai $$dy={\frac {\partial y}{\partial x_{1}}}\Delta x_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}\Delta x_{n}.$$

Karena, berdasarkan definisi,

$dx_{i}(\Delta x_{1},\dots ,\Delta x_{n})=\Delta x_{i},$

maka

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

Untuk kasus diferensial parsial dengan satu variabel, maka berlaku identitas aproksimasi berikut $$dy\approx \Delta y.$$ Pada identitas tersebut, galat totalnya dapat dibuat sekecil-kecil mungkin dengan ${\textstyle {\sqrt {\Delta x_{1}^{2}+\cdots +\Delta x_{n}^{2}}}}$. Hal tersebut dilakukan dengan membatasi pertambahan menjadi bernilai kecil.

Penerapan diferensial total untuk mengestimasi galat

In measurement, the total differential is used in estimating the error $\Delta f$ of a function $f$ 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 $x$ gives the sensitivity of the function $f$ to a change in $x$, 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 $f(a,b)=ab$;

$\Delta f=f_{a}\Delta a+f_{b}\Delta b$; evaluating the derivatives

Δf = bΔa + aΔb; dividing by f, which is a × b

Δf/f = Δa/a + Δb/b

That is to say, in multiplication, the total relative 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

Δf/f = Δa/a + Δb/(b ln b)

with an extra 'ln b' factor not found in the case of a simple product. This additional factor tends to make the error smaller, as ln b is not as large as a bare b.

Diferemsial tingkat tinggi

Higher-order differentials of a function y = f(x) of a single variable x can be defined via:

$d^{2}y=d(dy)=d(f'(x)dx)=(df'(x))dx=f''(x)\,(dx)^{2},$

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 variable x itself is permitted to depend on other variables, then the expression becomes more complicated, as it must include also higher order differentials in x itself. Thus, for instance,

${\begin{aligned}d^{2}y&=f''(x)\,(dx)^{2}+f'(x)d^{2}x\\d^{3}y&=f'''(x)\,(dx)^{3}+3f''(x)dx\,d^{2}x+f'(x)d^{3}x\end{aligned}}$

and so forth.

Similar considerations apply to defining higher order differentials of functions of several variables. For example, if f is a function of two variables x and y, 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 a binomial coefficient. In more variables, an analogous expression holds, but with an appropriate multinomial expansion rather than binomial expansion.

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 function f of x and y 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 infelicity, the use of higher order differentials was roundly criticized by Hadamard 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.

In these contexts, the nth order differential of the function f applied to an increment Δx 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 an nth forward difference with increment tΔx.

This definition makes sense as well if f is a function of several variables (for simplicity taken here as a vector argument). Then the nth differential defined in this way is a homogeneous function of degree n in the vector increment Δx. Furthermore, the Taylor series of f at the point x 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 order Gateaux derivative generalizes these considerations to infinite dimensional spaces.

Sifat-sifat

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:

• Linearity: For constants a and b and differentiable functions f and g,

$d(af+bg)=a\,df+b\,dg.$

• Product rule: For two differentiable functions f and g,

$d(fg)=f\,dg+g\,df.$

An operation d with these two properties is known in abstract algebra as a derivation. They imply the Power rule

$d(f^{n})=nf^{n-1}df$

In addition, various forms of the chain rule hold, in increasing level of generality:

• If y = f(u) is a differentiable function of the variable u and u = g(x) is a differentiable function of x, then

$dy=f'(u)\,du=f'(g(x))g'(x)\,dx.$

• If y = f(x₁, ..., x_n) and all of the variables x₁, ..., x_n depend on another variable t, then by the chain rule for partial derivatives, one has

${\begin{aligned}dy&={\frac {dy}{dt}}dt\\&={\frac {\partial y}{\partial x_{1}}}dx_{1}+\cdots +{\frac {\partial y}{\partial x_{n}}}dx_{n}\\&={\frac {\partial y}{\partial x_{1}}}{\frac {dx_{1}}{dt}}\,dt+\cdots +{\frac {\partial y}{\partial x_{n}}}{\frac {dx_{n}}{dt}}\,dt.\end{aligned}}$

Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantity dt.

• More general analogous expressions hold, in which the intermediate variables x_i depend on more than one variable.

Perumusan umum

Lihat pula: Fréchet derivative dan Gateaux derivative

A consistent notion of differential can be developed for a function f : Rⁿ → R^m between two Euclidean spaces. Let x,Δx ∈ Rⁿ be a pair of Euclidean vectors. The increment in the function f is

$\Delta f=f(\mathbf {x} +\Delta \mathbf {x} )-f(\mathbf {x} ).$

If there exists an m × n matrix A such that

$\Delta f=A\Delta \mathbf {x} +\|\Delta \mathbf {x} \|{\boldsymbol {\varepsilon }}$

in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix, and the linear transformation that associates to the increment Δx ∈ Rⁿ the vector AΔx ∈ R^m is, in this general setting, known as the differential df(x) of f at the point x. This is precisely the Fréchet derivative, and the same construction can be made to work for a function between any Banach spaces.

Another fruitful point of view is to define the differential directly as a kind of directional 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). If t represents time and x position, then h 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 the tangent space, and so df gives a linear function on the tangent space: a differential form. With this interpretation, the differential of f is known as the exterior derivative, and has broad application in differential geometry because the notion of velocities and the tangent space makes sense on any differentiable manifold. If, in addition, the output value of f also represents a position (in a Euclidean space), then a dimensional analysis confirms that the output value of df must be a velocity. If one treats the differential in this manner, then it is known as the pushforward since it "pushes" velocities from a source space into velocities in a target space.

Pendekatan lain

Artikel utama: Differential (infinitesimal)

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

• Defining the differential as a kind of differential form, specifically the exterior derivative of a function. The infinitesimal increments are then identified with vectors in the tangent space at a point. This approach is popular in differential geometry and related fields, because it readily generalizes to mappings between differentiable manifolds.
• Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry.
• Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced.
• Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers which contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson.

Contoh dan penerapan

Differentials may be effectively used in numerical analysis to study the propagation of experimental errors in a calculation, and thus the overall numerical stability of a problem (Courant 1937a). Suppose that the variable x represents the outcome of an experiment and y is the result of a numerical computation applied to x. The question is to what extent errors in the measurement of x influence the outcome of the computation of y. If the x is known to within Δx of its true value, then Taylor's theorem gives the following estimate on the error Δy in the computation of y:

$\Delta y=f'(x)\Delta x+{\frac {(\Delta x)^{2}}{2}}f''(\xi )$

where ξ = x + θΔx for some 0 < θ < 1. If Δx is small, then the second order term is negligible, so that Δy is, for practical purposes, well-approximated by dy = f'(x)Δx.

The differential is often useful to rewrite a differential equation

${\frac {dy}{dx}}=g(x)$

in the form

$dy=g(x)\,dx,$

in particular when one wants to separate the variables.

See also

• Notation for differentiation

References

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External links

• Differential Of A Function at Wolfram Demonstrations Project