Difference quotient

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class="vector-toc-numb">2</span> <span>Defining the point range</span> </div> </a> <ul id="toc-Defining_the_point_range-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_primary_difference_quotient_(Ń_=_1)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_primary_difference_quotient_(Ń_=_1)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The primary difference quotient (<i>Ń</i> = 1)</span> </div> </a> <button aria-controls="toc-The_primary_difference_quotient_(Ń_=_1)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle The primary difference quotient (<i>Ń</i> = 1) subsection</span> </button> <ul id="toc-The_primary_difference_quotient_(Ń_=_1)-sublist" class="vector-toc-list"> <li id="toc-As_a_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>As a derivative</span> </div> </a> <ul id="toc-As_a_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_divided_difference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_divided_difference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>As a divided difference</span> </div> </a> <ul id="toc-As_a_divided_difference-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Higher-order_difference_quotients" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_difference_quotients"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Higher-order difference quotients</span> </div> </a> <button aria-controls="toc-Higher-order_difference_quotients-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Higher-order difference quotients subsection</span> </button> <ul id="toc-Higher-order_difference_quotients-sublist" class="vector-toc-list"> <li id="toc-Second_order" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Second order</span> </div> </a> <ul id="toc-Second_order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Third_order" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Third_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Third order</span> </div> </a> <ul id="toc-Third_order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nth_order" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nth_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span><i>N</i>th order</span> </div> </a> <ul id="toc-Nth_order-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applying_the_divided_difference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applying_the_divided_difference"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applying the divided difference</span> </div> </a> <ul id="toc-Applying_the_divided_difference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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href="https://it.wikipedia.org/wiki/Rapporto_incrementale" title="Rapporto incrementale – Italian" lang="it" hreflang="it" data-title="Rapporto incrementale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Differentiequoti%C3%ABnt" title="Differentiequotiënt – Dutch" lang="nl" hreflang="nl" data-title="Differentiequotiënt" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B7%AE%E5%88%86%E5%95%86" title="差分商 – Japanese" lang="ja" hreflang="ja" data-title="差分商" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Iloraz_r%C3%B3%C5%BCnicowy" title="Iloraz różnicowy – Polish" lang="pl" hreflang="pl" data-title="Iloraz różnicowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Coeficiente_diferencial" title="Coeficiente diferencial – Portuguese" lang="pt" hreflang="pt" data-title="Coeficiente diferencial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Difference_quotient" title="Difference quotient – Simple English" lang="en-simple" hreflang="en-simple" data-title="Difference quotient" data-language-autonym="Simple English" data-language-local-name="Simple English" 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.hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a>.</div> <p>In single-variable <a href="/wiki/Calculus" title="Calculus">calculus</a>, the <b>difference quotient</b> is usually the name for the expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(x+h)-f(x)}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(x+h)-f(x)}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68a16d01cd0c630cd3ca58299e941eaa9b218e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.691ex; height:5.843ex;" alt="{\displaystyle {\frac {f(x+h)-f(x)}{h}}}"></span></dd></dl> <p>which when taken to the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a> as <i>h</i> approaches 0 gives the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i>.<sup id="cite_ref-LaxTerrell2013_1-0" class="reference"><a href="#cite_note-LaxTerrell2013-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HockettBock2005_2-0" class="reference"><a href="#cite_note-HockettBock2005-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ryan2010_3-0" class="reference"><a href="#cite_note-Ryan2010-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-NealGustafson2012_4-0" class="reference"><a href="#cite_note-NealGustafson2012-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The name of the expression stems from the fact that it is the <a href="/wiki/Quotient" title="Quotient">quotient</a> of the <a href="/wiki/Difference_(mathematics)" class="mw-redirect" title="Difference (mathematics)">difference</a> of values of the function by the difference of the corresponding values of its argument (the latter is (<i>x</i> + <i>h</i>) - <i>x</i> = <i>h</i> in this case).<sup id="cite_ref-Comenetz2002_5-0" class="reference"><a href="#cite_note-Comenetz2002-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pasch2010_6-0" class="reference"><a href="#cite_note-Pasch2010-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The difference quotient is a measure of the <a href="/wiki/Average" title="Average">average</a> <a href="/wiki/Rate_of_change_(mathematics)" class="mw-redirect" title="Rate of change (mathematics)">rate of change</a> of the function over an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> (in this case, an interval of length <i>h</i>).<sup id="cite_ref-WilsonAdamson2008_7-0" class="reference"><a href="#cite_note-WilsonAdamson2008-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RubySellers2014_8-0" class="reference"><a href="#cite_note-RubySellers2014-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 237">: 237 </span></sup><sup id="cite_ref-HungerfordShaw2008_9-0" class="reference"><a href="#cite_note-HungerfordShaw2008-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The limit of the difference quotient (i.e., the derivative) is thus the <a href="/wiki/Instantaneous" class="mw-redirect" title="Instantaneous">instantaneous</a> rate of change.<sup id="cite_ref-HungerfordShaw2008_9-1" class="reference"><a href="#cite_note-HungerfordShaw2008-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>By a slight change in notation (and viewpoint), for an interval [<i>a</i>, <i>b</i>], the difference quotient </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(b)-f(a)}{b-a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(b)-f(a)}{b-a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fed83c0443dce2a8d4c231747a1d33f72b239fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.08ex; height:6.009ex;" alt="{\displaystyle {\frac {f(b)-f(a)}{b-a}}}"></span></dd></dl> <p>is called<sup id="cite_ref-Comenetz2002_5-1" class="reference"><a href="#cite_note-Comenetz2002-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> the mean (or average) value of the derivative of <i>f</i> over the interval [<i>a</i>, <i>b</i>]. This name is justified by the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>, which states that for a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> <i>f</i>, its derivative <i>f<span class="nowrap" style="padding-left:0.15em;">′</span></i> reaches its <a href="/wiki/Mean_of_a_function" title="Mean of a function">mean value</a> at some point in the interval.<sup id="cite_ref-Comenetz2002_5-2" class="reference"><a href="#cite_note-Comenetz2002-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Geometrically, this difference quotient measures the <a href="/wiki/Slope" title="Slope">slope</a> of the <a href="/wiki/Secant_line" title="Secant line">secant line</a> passing through the points with coordinates (<i>a</i>, <i>f</i>(<i>a</i>)) and (<i>b</i>, <i>f</i>(<i>b</i>)).<sup id="cite_ref-Krantz2014_10-0" class="reference"><a href="#cite_note-Krantz2014-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Difference quotients are used as approximations in <a href="/wiki/Numerical_differentiation" title="Numerical differentiation">numerical differentiation</a>,<sup id="cite_ref-RubySellers2014_8-1" class="reference"><a href="#cite_note-RubySellers2014-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> but they have also been subject of criticism in this application.<sup id="cite_ref-GriewankWalther2008_11-0" class="reference"><a href="#cite_note-GriewankWalther2008-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Difference quotients may also find relevance in applications involving <a href="/wiki/Temporal_discretization" title="Temporal discretization">Time discretization</a>, where the width of the time step is used for the value of h. </p><p>The difference quotient is sometimes also called the <b>Newton quotient</b><sup id="cite_ref-Krantz2014_10-1" class="reference"><a href="#cite_note-Krantz2014-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lang1968_12-0" class="reference"><a href="#cite_note-Lang1968-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hahn1994_13-0" class="reference"><a href="#cite_note-Hahn1994-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ClaphamNicholson2009_14-0" class="reference"><a href="#cite_note-ClaphamNicholson2009-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> (after <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>) or <b>Fermat's difference quotient</b> (after <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The typical notion of the difference quotient discussed above is a particular case of a more general concept. The primary vehicle of <a href="/wiki/Calculus" title="Calculus">calculus</a> and other higher mathematics is the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>. Its "input value" is its <i>argument</i>, usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their <a href="/wiki/Delta_(letter)" title="Delta (letter)">Delta</a> (Δ<i>P</i>), as is the difference in their function result, the particular notation being determined by the direction of formation: </p> <ul><li>Forward difference: Δ<i>F</i>(<i>P</i>) = <i>F</i>(<i>P</i> + Δ<i>P</i>) − <i>F</i>(<i>P</i>);</li> <li>Central difference: δF(P) = F(P + <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>ΔP) − F(P − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>ΔP);</li> <li>Backward difference: ∇F(P) = F(P) − F(P − ΔP).</li></ul> <p>The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore, </p> <ul><li>If |ΔP| is <i>finite</i> (meaning measurable), then ΔF(P) is known as a <b><a href="/wiki/Finite_difference" title="Finite difference">finite difference</a></b>, with specific denotations of DP and DF(P);</li> <li>If |ΔP| is <i><a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a></i> (an infinitely small amount—<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ι</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iota }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce48dd56254d0a7c33e987c7c8eeb44c963ac04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.823ex; height:1.676ex;" alt="{\displaystyle \iota }"></span></i>—usually expressed in standard analysis as a limit: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\Delta P\rightarrow 0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\Delta P\rightarrow 0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3415178067afd16d952c9f31dfe17586f33fe4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-right: -0.387ex; width:5.455ex; height:4.009ex;" alt="{\displaystyle \lim _{\Delta P\rightarrow 0}\,\!}"></span>), then ΔF(P) is known as an <b>infinitesimal difference</b>, with specific denotations of dP and dF(P) (in calculus graphing, the point is almost exclusively identified as "x" and F(x) as "y").</li></ul> <p>The function difference divided by the point difference is known as "difference quotient": </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta F(P)}{\Delta P}}={\frac {F(P+\Delta P)-F(P)}{\Delta P}}={\frac {\nabla F(P+\Delta P)}{\Delta P}}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta F(P)}{\Delta P}}={\frac {F(P+\Delta P)-F(P)}{\Delta P}}={\frac {\nabla F(P+\Delta P)}{\Delta P}}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/319fb59778b043e55bcbf01e9c71c01cc8efa5a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:50.677ex; height:5.843ex;" alt="{\displaystyle {\frac {\Delta F(P)}{\Delta P}}={\frac {F(P+\Delta P)-F(P)}{\Delta P}}={\frac {\nabla F(P+\Delta P)}{\Delta P}}.\,\!}"></span></dd></dl> <p>If ΔP is infinitesimal, then the difference quotient is a <i><a href="/wiki/Derivative" title="Derivative">derivative</a></i>, otherwise it is a <i><a href="/wiki/Divided_differences" title="Divided differences">divided difference</a></i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{If }}|\Delta P|={\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {dF(P)}{dP}}=F'(P)=G(P);\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </mrow> <mo>:</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{If }}|\Delta P|={\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {dF(P)}{dP}}=F'(P)=G(P);\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d6ff970a5839b5ee2bb7b2c55bd06f87fb68ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:52.469ex; height:5.843ex;" alt="{\displaystyle {\text{If }}|\Delta P|={\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {dF(P)}{dP}}=F'(P)=G(P);\,\!}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{If }}|\Delta P|>{\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {DF(P)}{DP}}=F[P,P+\Delta P].\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi>ι</mi> </mrow> </mrow> <mo>:</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>P</mi> <mo>,</mo> <mi>P</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo stretchy="false">]</mo> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{If }}|\Delta P|>{\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {DF(P)}{DP}}=F[P,P+\Delta P].\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a9ce0612b2a5b0e4c2e0866c35c11b2e64c811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:52.724ex; height:5.843ex;" alt="{\displaystyle {\text{If }}|\Delta P|>{\mathit {\iota }}:\quad {\frac {\Delta F(P)}{\Delta P}}={\frac {DF(P)}{DP}}=F[P,P+\Delta P].\,\!}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Defining_the_point_range">Defining the point range</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=2" title="Edit section: Defining the point range"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (0.5) ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)): </p> <dl><dd>LB = Lower Boundary; UB = Upper Boundary;</dd></dl> <p>Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequential degrees ("higher orders") of derivation, or <i>differentiation</i>. This property can be generalized to all difference quotients.<br /> As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point (<i>P</i><sub><i>i</i></sub>), where LB = <i>P</i><sub>0</sub> and UB = <i>P</i><sub><i>ń</i></sub>, the <i>n</i>th point, equaling the degree/order: </p> <pre> LB = P<sub>0</sub> = P<sub>0</sub> + 0Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-0)Δ<sub>1</sub>P; P<sub>1</sub> = P<sub>0</sub> + 1Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-1)Δ<sub>1</sub>P; P<sub>2</sub> = P<sub>0</sub> + 2Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-2)Δ<sub>1</sub>P; P<sub>3</sub> = P<sub>0</sub> + 3Δ<sub>1</sub>P = P<sub>ń</sub> − (Ń-3)Δ<sub>1</sub>P; ↓ ↓ ↓ ↓ P<sub>ń-3</sub> = P<sub>0</sub> + (Ń-3)Δ<sub>1</sub>P = P<sub>ń</sub> − 3Δ<sub>1</sub>P; P<sub>ń-2</sub> = P<sub>0</sub> + (Ń-2)Δ<sub>1</sub>P = P<sub>ń</sub> − 2Δ<sub>1</sub>P; P<sub>ń-1</sub> = P<sub>0</sub> + (Ń-1)Δ<sub>1</sub>P = P<sub>ń</sub> − 1Δ<sub>1</sub>P; UB = P<sub>ń-0</sub> = P<sub>0</sub> + (Ń-0)Δ<sub>1</sub>P = P<sub>ń</sub> − 0Δ<sub>1</sub>P = P<sub>ń</sub>; </pre> <pre> ΔP = Δ<sub>1</sub>P = P<sub>1</sub> − P<sub>0</sub> = P<sub>2</sub> − P<sub>1</sub> = P<sub>3</sub> − P<sub>2</sub> = ... = P<sub>ń</sub> − P<sub>ń-1</sub>; </pre> <pre> ΔB = UB − LB = P<sub>ń</sub> − P<sub>0</sub> = Δ<sub>ń</sub>P = ŃΔ<sub>1</sub>P. </pre> <div class="mw-heading mw-heading2"><h2 id="The_primary_difference_quotient_(Ń_=_1)"><span id="The_primary_difference_quotient_.28.C5.83_.3D_1.29"></span>The primary difference quotient (<i>Ń</i> = 1)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=3" title="Edit section: The primary difference quotient (Ń = 1)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta F(P_{0})}{\Delta P}}={\frac {F(P_{\acute {n}})-F(P_{0})}{\Delta _{\acute {n}}P}}={\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta F(P_{0})}{\Delta P}}={\frac {F(P_{\acute {n}})-F(P_{0})}{\Delta _{\acute {n}}P}}={\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef17de030b68d5436ee9a856611cefd8af85d080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-right: -0.387ex; width:66.971ex; height:6.843ex;" alt="{\displaystyle {\frac {\Delta F(P_{0})}{\Delta P}}={\frac {F(P_{\acute {n}})-F(P_{0})}{\Delta _{\acute {n}}P}}={\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}.\,\!}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="As_a_derivative">As a derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=4" title="Edit section: As a derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd>The difference quotient as a derivative needs no explanation, other than to point out that, since P<sub>0</sub> essentially equals P<sub>1</sub> = P<sub>2</sub> = ... = P<sub>ń</sub> (as the differences are infinitesimal), the <a href="/wiki/Leibniz_notation" class="mw-redirect" title="Leibniz notation">Leibniz notation</a> and derivative expressions do not distinguish P to P<sub>0</sub> or P<sub>ń</sub>:</dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dF(P)}{dP}}={\frac {F(P_{1})-F(P_{0})}{dP}}=F'(P)=G(P).\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dF(P)}{dP}}={\frac {F(P_{1})-F(P_{0})}{dP}}=F'(P)=G(P).\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f52a9fe0b696b36c9672870b4605ca6903008f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:44.982ex; height:5.843ex;" alt="{\displaystyle {\frac {dF(P)}{dP}}={\frac {F(P_{1})-F(P_{0})}{dP}}=F'(P)=G(P).\,\!}"></span></dd></dl></dd></dl></dd></dl> <p>There are <a href="/wiki/Derivative#Notation_for_differentiation" title="Derivative">other derivative notations</a>, but these are the most recognized, standard designations. </p> <div class="mw-heading mw-heading3"><h3 id="As_a_divided_difference">As a divided difference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=5" title="Edit section: As a divided difference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd>A divided difference, however, does require further elucidation, as it equals the average derivative between and including LB and UB:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P_{(tn)}&=LB+{\frac {TN-1}{UT-1}}\Delta B\ =UB-{\frac {UT-TN}{UT-1}}\Delta B;\\[10pt]&{}\qquad {\color {white}.}(P_{(1)}=LB,\ P_{(ut)}=UB){\color {white}.}\\[10pt]F'(P_{\tilde {a}})&=F'(LB<P<UB)=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <mi>B</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>U</mi> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mtext> </mtext> <mo>=</mo> <mi>U</mi> <mi>B</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <mi>T</mi> <mo>−</mo> <mi>T</mi> <mi>N</mi> </mrow> <mrow> <mi>U</mi> <mi>T</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mo>;</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mo>.</mo> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>L</mi> <mi>B</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>u</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mo>.</mo> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>T</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{(tn)}&=LB+{\frac {TN-1}{UT-1}}\Delta B\ =UB-{\frac {UT-TN}{UT-1}}\Delta B;\\[10pt]&{}\qquad {\color {white}.}(P_{(1)}=LB,\ P_{(ut)}=UB){\color {white}.}\\[10pt]F'(P_{\tilde {a}})&=F'(LB<P<UB)=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f856f510e32d4991b600e7822810ac090e2f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.694ex; margin-bottom: -0.311ex; width:54.387ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}P_{(tn)}&=LB+{\frac {TN-1}{UT-1}}\Delta B\ =UB-{\frac {UT-TN}{UT-1}}\Delta B;\\[10pt]&{}\qquad {\color {white}.}(P_{(1)}=LB,\ P_{(ut)}=UB){\color {white}.}\\[10pt]F'(P_{\tilde {a}})&=F'(LB<P<UB)=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}}.\end{aligned}}}"></span></dd></dl></dd></dl> <dl><dd>In this interpretation, P<sub>ã</sub> represents a function extracted, average value of P (midrange, but usually not exactly midpoint), the particular valuation depending on the function averaging it is extracted from. More formally, P<sub>ã</sub> is found in the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a> of calculus, which says:</dd></dl> <dl><dd><dl><dd><i>For any function that is continuous on [LB,UB] and differentiable on (LB,UB) there exists some P<sub>ã</sub> in the interval (LB,UB) such that the secant joining the endpoints of the interval [LB,UB] is parallel to the tangent at P<sub>ã</sub>.</i></dd></dl></dd></dl> <dl><dd>Essentially, P<sub>ã</sub> denotes some value of P between LB and UB—hence,</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\tilde {a}}:=LB<P<UB=P_{0}<P<P_{\acute {n}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </msub> <mo>:=</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\tilde {a}}:=LB<P<UB=P_{0}<P<P_{\acute {n}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaf76e3f5d4fda267813abc7857eee79dcd045f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:37.86ex; height:2.843ex;" alt="{\displaystyle P_{\tilde {a}}:=LB<P<UB=P_{0}<P<P_{\acute {n}}\,\!}"></span></dd></dl></dd></dl> <dl><dd>which links the mean value result with the divided difference:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {DF(P_{0})}{DP}}&=F[P_{0},P_{1}]={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}=F'(P_{0}<P<P_{1})=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}},\\[8pt]&={\frac {DF(LB)}{DB}}={\frac {\Delta F(LB)}{\Delta B}}={\frac {\nabla F(UB)}{\Delta B}},\\[8pt]&=F[LB,UB]={\frac {F(UB)-F(LB)}{UB-LB}},\\[8pt]&=F'(LB<P<UB)=G(LB<P<UB).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <mi>P</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>T</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>L</mi> <mi>B</mi> <mo>,</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>B</mi> <mo>−</mo> <mi>L</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {DF(P_{0})}{DP}}&=F[P_{0},P_{1}]={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}=F'(P_{0}<P<P_{1})=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}},\\[8pt]&={\frac {DF(LB)}{DB}}={\frac {\Delta F(LB)}{\Delta B}}={\frac {\nabla F(UB)}{\Delta B}},\\[8pt]&=F[LB,UB]={\frac {F(UB)-F(LB)}{UB-LB}},\\[8pt]&=F'(LB<P<UB)=G(LB<P<UB).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d11d71d39b0cac5f743df7f793b5423a220627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.505ex; width:80.479ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {DF(P_{0})}{DP}}&=F[P_{0},P_{1}]={\frac {F(P_{1})-F(P_{0})}{P_{1}-P_{0}}}=F'(P_{0}<P<P_{1})=\sum _{TN=1}^{UT=\infty }{\frac {F'(P_{(tn)})}{UT}},\\[8pt]&={\frac {DF(LB)}{DB}}={\frac {\Delta F(LB)}{\Delta B}}={\frac {\nabla F(UB)}{\Delta B}},\\[8pt]&=F[LB,UB]={\frac {F(UB)-F(LB)}{UB-LB}},\\[8pt]&=F'(LB<P<UB)=G(LB<P<UB).\end{aligned}}}"></span></dd></dl></dd></dl> <dl><dd>As there is, by its very definition, a tangible difference between LB/P<sub>0</sub> and UB/P<sub>ń</sub>, the Leibniz and derivative expressions <i>do</i> require <a href="/wiki/Divaricate" title="Divaricate">divarication</a> of the function argument.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Higher-order_difference_quotients">Higher-order difference quotients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=6" title="Edit section: Higher-order difference quotients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Second_order">Second order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=7" title="Edit section: Second order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\Delta ^{2}F(P_{0})}{\Delta _{1}P^{2}}}&={\frac {\Delta F'(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{2})-F(P_{1})}{\Delta _{1}P}}-{\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}};\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\Delta ^{2}F(P_{0})}{\Delta _{1}P^{2}}}&={\frac {\Delta F'(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{2})-F(P_{1})}{\Delta _{1}P}}-{\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}};\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7653e6ebfe8b3047239f1f8ceca7a6b11a4c706d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.838ex; width:44.026ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\Delta ^{2}F(P_{0})}{\Delta _{1}P^{2}}}&={\frac {\Delta F'(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{2})-F(P_{1})}{\Delta _{1}P}}-{\frac {F(P_{1})-F(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}};\end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d^{2}F(P)}{dP^{2}}}&={\frac {dF'(P)}{dP}}={\frac {F'(P_{1})-F'(P_{0})}{dP}},\\[10pt]&=\ {\frac {dG(P)}{dP}}={\frac {G(P_{1})-G(P_{0})}{dP}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{dP^{2}}},\\[10pt]&=F''(P)=G'(P)=H(P)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>G</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d^{2}F(P)}{dP^{2}}}&={\frac {dF'(P)}{dP}}={\frac {F'(P_{1})-F'(P_{0})}{dP}},\\[10pt]&=\ {\frac {dG(P)}{dP}}={\frac {G(P_{1})-G(P_{0})}{dP}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{dP^{2}}},\\[10pt]&=F''(P)=G'(P)=H(P)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4c034760e8831db35c8be43466dca9801cc899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.505ex; width:41.492ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d^{2}F(P)}{dP^{2}}}&={\frac {dF'(P)}{dP}}={\frac {F'(P_{1})-F'(P_{0})}{dP}},\\[10pt]&=\ {\frac {dG(P)}{dP}}={\frac {G(P_{1})-G(P_{0})}{dP}},\\[10pt]&={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{dP^{2}}},\\[10pt]&=F''(P)=G'(P)=H(P)\end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {D^{2}F(P_{0})}{DP^{2}}}&={\frac {DF'(P_{0})}{DP}}={\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \neq {\frac {F'(P_{1})-F'(P_{0})}{P_{1}-P_{0}}},\\[10pt]&=F[P_{0},P_{1},P_{2}]={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F''(P_{0}<P<P_{2})=\sum _{TN=1}^{\infty }{\frac {F''(P_{(tn)})}{UT}},\\[10pt]&=G'(P_{0}<P<P_{2})=H(P_{0}<P<P_{2}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mo>.</mo> </mstyle> </mrow> <mspace width="2em" /> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {D^{2}F(P_{0})}{DP^{2}}}&={\frac {DF'(P_{0})}{DP}}={\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \neq {\frac {F'(P_{1})-F'(P_{0})}{P_{1}-P_{0}}},\\[10pt]&=F[P_{0},P_{1},P_{2}]={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F''(P_{0}<P<P_{2})=\sum _{TN=1}^{\infty }{\frac {F''(P_{(tn)})}{UT}},\\[10pt]&=G'(P_{0}<P<P_{2})=H(P_{0}<P<P_{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd944930910885fd11b60397bc2fce4f2bf801f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.618ex; margin-bottom: -0.22ex; width:65.487ex; height:38.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {D^{2}F(P_{0})}{DP^{2}}}&={\frac {DF'(P_{0})}{DP}}={\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \neq {\frac {F'(P_{1})-F'(P_{0})}{P_{1}-P_{0}}},\\[10pt]&=F[P_{0},P_{1},P_{2}]={\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F''(P_{0}<P<P_{2})=\sum _{TN=1}^{\infty }{\frac {F''(P_{(tn)})}{UT}},\\[10pt]&=G'(P_{0}<P<P_{2})=H(P_{0}<P<P_{2}).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Third_order">Third order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=8" title="Edit section: Third order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\Delta ^{3}F(P_{0})}{\Delta _{1}P^{3}}}&={\frac {\Delta ^{2}F'(P_{0})}{\Delta _{1}P^{2}}}={\frac {\Delta F''(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {{\frac {\Delta F(P_{2})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}-{\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{3})-2F(P_{2})+F(P_{1})}{\Delta _{1}P^{2}}}-{\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{\Delta _{1}P^{3}}};\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\Delta ^{3}F(P_{0})}{\Delta _{1}P^{3}}}&={\frac {\Delta ^{2}F'(P_{0})}{\Delta _{1}P^{2}}}={\frac {\Delta F''(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {{\frac {\Delta F(P_{2})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}-{\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{3})-2F(P_{2})+F(P_{1})}{\Delta _{1}P^{2}}}-{\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{\Delta _{1}P^{3}}};\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ef9fc1aa01c46effe4ddf371f9b183da1d1dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.505ex; width:59.426ex; height:40.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\Delta ^{3}F(P_{0})}{\Delta _{1}P^{3}}}&={\frac {\Delta ^{2}F'(P_{0})}{\Delta _{1}P^{2}}}={\frac {\Delta F''(P_{0})}{\Delta _{1}P}}={\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {{\frac {\Delta F(P_{2})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}-{\frac {{\frac {\Delta F'(P_{1})}{\Delta _{1}P}}-{\frac {\Delta F'(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F(P_{3})-2F(P_{2})+F(P_{1})}{\Delta _{1}P^{2}}}-{\frac {F(P_{2})-2F(P_{1})+F(P_{0})}{\Delta _{1}P^{2}}}}{\Delta _{1}P}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{\Delta _{1}P^{3}}};\end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d^{3}F(P)}{dP^{3}}}&={\frac {d^{2}F'(P)}{dP^{2}}}={\frac {dF''(P)}{dP}}={\frac {F''(P_{1})-F''(P_{0})}{dP}},\\[10pt]&={\frac {d^{2}G(P)}{dP^{2}}}\ ={\frac {dG'(P)}{dP}}\ ={\frac {G'(P_{1})-G'(P_{0})}{dP}},\\[10pt]&{\color {white}.}\qquad \qquad \ \ ={\frac {dH(P)}{dP}}\ ={\frac {H(P_{1})-H(P_{0})}{dP}},\\[10pt]&={\frac {G(P_{2})-2G(P_{1})+G(P_{0})}{dP^{2}}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{dP^{3}}},\\[10pt]&=F'''(P)=G''(P)=H'(P)=I(P);\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mo>.</mo> </mstyle> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mtext> </mtext> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>G</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d^{3}F(P)}{dP^{3}}}&={\frac {d^{2}F'(P)}{dP^{2}}}={\frac {dF''(P)}{dP}}={\frac {F''(P_{1})-F''(P_{0})}{dP}},\\[10pt]&={\frac {d^{2}G(P)}{dP^{2}}}\ ={\frac {dG'(P)}{dP}}\ ={\frac {G'(P_{1})-G'(P_{0})}{dP}},\\[10pt]&{\color {white}.}\qquad \qquad \ \ ={\frac {dH(P)}{dP}}\ ={\frac {H(P_{1})-H(P_{0})}{dP}},\\[10pt]&={\frac {G(P_{2})-2G(P_{1})+G(P_{0})}{dP^{2}}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{dP^{3}}},\\[10pt]&=F'''(P)=G''(P)=H'(P)=I(P);\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f3dd2ad0b83746c1a0246f73bc1f0927b75619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.005ex; width:55.111ex; height:45.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d^{3}F(P)}{dP^{3}}}&={\frac {d^{2}F'(P)}{dP^{2}}}={\frac {dF''(P)}{dP}}={\frac {F''(P_{1})-F''(P_{0})}{dP}},\\[10pt]&={\frac {d^{2}G(P)}{dP^{2}}}\ ={\frac {dG'(P)}{dP}}\ ={\frac {G'(P_{1})-G'(P_{0})}{dP}},\\[10pt]&{\color {white}.}\qquad \qquad \ \ ={\frac {dH(P)}{dP}}\ ={\frac {H(P_{1})-H(P_{0})}{dP}},\\[10pt]&={\frac {G(P_{2})-2G(P_{1})+G(P_{0})}{dP^{2}}},\\[10pt]&={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{dP^{3}}},\\[10pt]&=F'''(P)=G''(P)=H'(P)=I(P);\end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {D^{3}F(P_{0})}{DP^{3}}}&={\frac {D^{2}F'(P_{0})}{DP^{2}}}={\frac {DF''(P_{0})}{DP}}={\frac {F''(P_{1}<P<P_{3})-F''(P_{0}<P<P_{2})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \ \ \neq {\frac {F''(P_{1})-F''(P_{0})}{P_{1}-P_{0}}},\\[10pt]&={\frac {{\frac {F'(P_{2}<P<P_{3})-F'(P_{1}<P<P_{2})}{P_{1}-P_{0}}}-{\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}}}{P_{1}-P_{0}}},\\[10pt]&={\frac {F'(P_{2}<P<P_{3})-2F'(P_{1}<P<P_{2})+F'(P_{0}<P<P_{1})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F[P_{0},P_{1},P_{2},P_{3}]={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{(P_{1}-P_{0})^{3}}},\\[10pt]&=F'''(P_{0}<P<P_{3})=\sum _{TN=1}^{UT=\infty }{\frac {F'''(P_{(tn)})}{UT}},\\[10pt]&=G''(P_{0}<P<P_{3})\ =H'(P_{0}<P<P_{3})=I(P_{0}<P<P_{3}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow 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class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo 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<mo>‴</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>G</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {D^{3}F(P_{0})}{DP^{3}}}&={\frac {D^{2}F'(P_{0})}{DP^{2}}}={\frac {DF''(P_{0})}{DP}}={\frac {F''(P_{1}<P<P_{3})-F''(P_{0}<P<P_{2})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \ \ \neq {\frac {F''(P_{1})-F''(P_{0})}{P_{1}-P_{0}}},\\[10pt]&={\frac {{\frac {F'(P_{2}<P<P_{3})-F'(P_{1}<P<P_{2})}{P_{1}-P_{0}}}-{\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}}}{P_{1}-P_{0}}},\\[10pt]&={\frac {F'(P_{2}<P<P_{3})-2F'(P_{1}<P<P_{2})+F'(P_{0}<P<P_{1})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F[P_{0},P_{1},P_{2},P_{3}]={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{(P_{1}-P_{0})^{3}}},\\[10pt]&=F'''(P_{0}<P<P_{3})=\sum _{TN=1}^{UT=\infty }{\frac {F'''(P_{(tn)})}{UT}},\\[10pt]&=G''(P_{0}<P<P_{3})\ =H'(P_{0}<P<P_{3})=I(P_{0}<P<P_{3}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ba44a0f50679c97f5b387af7dd11daf4d39ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -28.403ex; margin-bottom: -0.269ex; width:80.613ex; height:58.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {D^{3}F(P_{0})}{DP^{3}}}&={\frac {D^{2}F'(P_{0})}{DP^{2}}}={\frac {DF''(P_{0})}{DP}}={\frac {F''(P_{1}<P<P_{3})-F''(P_{0}<P<P_{2})}{P_{1}-P_{0}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \ \ \neq {\frac {F''(P_{1})-F''(P_{0})}{P_{1}-P_{0}}},\\[10pt]&={\frac {{\frac {F'(P_{2}<P<P_{3})-F'(P_{1}<P<P_{2})}{P_{1}-P_{0}}}-{\frac {F'(P_{1}<P<P_{2})-F'(P_{0}<P<P_{1})}{P_{1}-P_{0}}}}{P_{1}-P_{0}}},\\[10pt]&={\frac {F'(P_{2}<P<P_{3})-2F'(P_{1}<P<P_{2})+F'(P_{0}<P<P_{1})}{(P_{1}-P_{0})^{2}}},\\[10pt]&=F[P_{0},P_{1},P_{2},P_{3}]={\frac {F(P_{3})-3F(P_{2})+3F(P_{1})-F(P_{0})}{(P_{1}-P_{0})^{3}}},\\[10pt]&=F'''(P_{0}<P<P_{3})=\sum _{TN=1}^{UT=\infty }{\frac {F'''(P_{(tn)})}{UT}},\\[10pt]&=G''(P_{0}<P<P_{3})\ =H'(P_{0}<P<P_{3})=I(P_{0}<P<P_{3}).\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Nth_order"><i>N</i>th order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=9" title="Edit section: Nth order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta ^{\acute {n}}F(P_{0})&=F^{({\acute {n}}-1)}(P_{1})-F^{({\acute {n}}-1)}(P_{0}),\\[10pt]&={\frac {F^{({\acute {n}}-2)}(P_{2})-F^{({\acute {n}}-2)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-2)}(P_{1})-F^{({\acute {n}}-2)}(P_{0})}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F^{({\acute {n}}-3)}(P_{3})-F^{({\acute {n}}-3)}(P_{2})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}\\[10pt]&{\color {white}.}\qquad -{\frac {{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{1})-F^{({\acute {n}}-3)}(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&=\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi 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</msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mo>.</mo> </mstyle> </mrow> <mspace width="2em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta ^{\acute {n}}F(P_{0})&=F^{({\acute {n}}-1)}(P_{1})-F^{({\acute {n}}-1)}(P_{0}),\\[10pt]&={\frac {F^{({\acute {n}}-2)}(P_{2})-F^{({\acute {n}}-2)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-2)}(P_{1})-F^{({\acute {n}}-2)}(P_{0})}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F^{({\acute {n}}-3)}(P_{3})-F^{({\acute {n}}-3)}(P_{2})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}\\[10pt]&{\color {white}.}\qquad -{\frac {{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{1})-F^{({\acute {n}}-3)}(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&=\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7dad583886237ae513ff67ad188c2030d3ea91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.838ex; width:67.017ex; height:40.843ex;" alt="{\displaystyle {\begin{aligned}\Delta ^{\acute {n}}F(P_{0})&=F^{({\acute {n}}-1)}(P_{1})-F^{({\acute {n}}-1)}(P_{0}),\\[10pt]&={\frac {F^{({\acute {n}}-2)}(P_{2})-F^{({\acute {n}}-2)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-2)}(P_{1})-F^{({\acute {n}}-2)}(P_{0})}{\Delta _{1}P}},\\[10pt]&={\frac {{\frac {F^{({\acute {n}}-3)}(P_{3})-F^{({\acute {n}}-3)}(P_{2})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}}{\Delta _{1}P}}\\[10pt]&{\color {white}.}\qquad -{\frac {{\frac {F^{({\acute {n}}-3)}(P_{2})-F^{({\acute {n}}-3)}(P_{1})}{\Delta _{1}P}}-{\frac {F^{({\acute {n}}-3)}(P_{1})-F^{({\acute {n}}-3)}(P_{0})}{\Delta _{1}P}}}{\Delta _{1}P}},\\[10pt]&=\cdots \end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\Delta ^{\acute {n}}F(P_{0})}{\Delta _{1}P^{\acute {n}}}}&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose {\acute {N}}-I}{{\acute {N}} \choose I}F(P_{0}+I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\\[10pt]&{\frac {\nabla ^{\acute {n}}F(P_{\acute {n}})}{\Delta _{1}P^{\acute {n}}}}\\[10pt]&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose I}{{\acute {N}} \choose I}F(P_{\acute {n}}-I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>´</mo> </mover> </mrow> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>I</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi>I</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>I</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>´</mo> </mover> </mrow> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mi>I</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>´</mo> </mover> </mrow> </mrow> <mi>I</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mo>−</mo> <mi>I</mi> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\Delta ^{\acute {n}}F(P_{0})}{\Delta _{1}P^{\acute {n}}}}&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose {\acute {N}}-I}{{\acute {N}} \choose I}F(P_{0}+I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\\[10pt]&{\frac {\nabla ^{\acute {n}}F(P_{\acute {n}})}{\Delta _{1}P^{\acute {n}}}}\\[10pt]&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose I}{{\acute {N}} \choose I}F(P_{\acute {n}}-I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9194c3bb135c82365145f40c4ad16c647502c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:45.552ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\Delta ^{\acute {n}}F(P_{0})}{\Delta _{1}P^{\acute {n}}}}&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose {\acute {N}}-I}{{\acute {N}} \choose I}F(P_{0}+I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\\[10pt]&{\frac {\nabla ^{\acute {n}}F(P_{\acute {n}})}{\Delta _{1}P^{\acute {n}}}}\\[10pt]&={\frac {\sum _{I=0}^{\acute {N}}{-1 \choose I}{{\acute {N}} \choose I}F(P_{\acute {n}}-I\Delta _{1}P)}{\Delta _{1}P^{\acute {n}}}};\end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d^{\acute {n}}F(P_{0})}{dP^{\acute {n}}}}&={\frac {d^{{\acute {n}}-1}F'(P_{0})}{dP^{{\acute {n}}-1}}}={\frac {d^{{\acute {n}}-2}F''(P_{0})}{dP^{{\acute {n}}-2}}}={\frac {d^{{\acute {n}}-3}F'''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}F^{(r)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&={\frac {d^{{\acute {n}}-1}G(P_{0})}{dP^{{\acute {n}}-1}}}\\[10pt]&={\frac {d^{{\acute {n}}-2}G'(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}G''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}G^{(r-1)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad ={\frac {d^{{\acute {n}}-2}H(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}H'(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}H^{(r-2)}(P_{0})}{dP^{{\acute {n}}-r}}},\\&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \qquad \ =\ {\frac {d^{{\acute {n}}-3}I(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}I^{(r-3)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&=F^{({\acute {n}})}(P)=G^{({\acute {n}}-1)}(P)=H^{({\acute {n}}-2)}(P)=I^{({\acute {n}}-3)}(P)=\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 0.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> 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class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> </mrow> </msup> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>r</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d^{\acute {n}}F(P_{0})}{dP^{\acute {n}}}}&={\frac {d^{{\acute {n}}-1}F'(P_{0})}{dP^{{\acute {n}}-1}}}={\frac {d^{{\acute {n}}-2}F''(P_{0})}{dP^{{\acute {n}}-2}}}={\frac {d^{{\acute {n}}-3}F'''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}F^{(r)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&={\frac {d^{{\acute {n}}-1}G(P_{0})}{dP^{{\acute {n}}-1}}}\\[10pt]&={\frac {d^{{\acute {n}}-2}G'(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}G''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}G^{(r-1)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad ={\frac {d^{{\acute {n}}-2}H(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}H'(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}H^{(r-2)}(P_{0})}{dP^{{\acute {n}}-r}}},\\&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \qquad \ =\ {\frac {d^{{\acute {n}}-3}I(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}I^{(r-3)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&=F^{({\acute {n}})}(P)=G^{({\acute {n}}-1)}(P)=H^{({\acute {n}}-2)}(P)=I^{({\acute {n}}-3)}(P)=\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd455f469966afb9e49acbfb7fd63550be318a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.657ex; margin-bottom: -0.181ex; width:81.463ex; height:48.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d^{\acute {n}}F(P_{0})}{dP^{\acute {n}}}}&={\frac {d^{{\acute {n}}-1}F'(P_{0})}{dP^{{\acute {n}}-1}}}={\frac {d^{{\acute {n}}-2}F''(P_{0})}{dP^{{\acute {n}}-2}}}={\frac {d^{{\acute {n}}-3}F'''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}F^{(r)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&={\frac {d^{{\acute {n}}-1}G(P_{0})}{dP^{{\acute {n}}-1}}}\\[10pt]&={\frac {d^{{\acute {n}}-2}G'(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}G''(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}G^{(r-1)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&{\color {white}.}\qquad \qquad \qquad ={\frac {d^{{\acute {n}}-2}H(P_{0})}{dP^{{\acute {n}}-2}}}=\ {\frac {d^{{\acute {n}}-3}H'(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}H^{(r-2)}(P_{0})}{dP^{{\acute {n}}-r}}},\\&{\color {white}.}\qquad \qquad \qquad \qquad \qquad \qquad \ =\ {\frac {d^{{\acute {n}}-3}I(P_{0})}{dP^{{\acute {n}}-3}}}=\cdots ={\frac {d^{{\acute {n}}-r}I^{(r-3)}(P_{0})}{dP^{{\acute {n}}-r}}},\\[10pt]&=F^{({\acute {n}})}(P)=G^{({\acute {n}}-1)}(P)=H^{({\acute {n}}-2)}(P)=I^{({\acute {n}}-3)}(P)=\cdots \end{aligned}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {D^{\acute {n}}F(P_{0})}{DP^{\acute {n}}}}&=F[P_{0},P_{1},P_{2},P_{3},\ldots ,P_{{\acute {n}}-3},P_{{\acute {n}}-2},P_{{\acute {n}}-1},P_{\acute {n}}],\\[10pt]&=F^{({\acute {n}})}(P_{0}<P<P_{\acute {n}})=\sum _{TN=1}^{UT=\infty }{\frac {F^{({\acute {n}})}(P_{(tn)})}{UT}}\\[10pt]&=F^{({\acute {n}})}(LB<P<UB)=G^{({\acute {n}}-1)}(LB<P<UB)=\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>D</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <mi>P</mi> <mo><</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>T</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo>´</mo> </mover> </mrow> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {D^{\acute {n}}F(P_{0})}{DP^{\acute {n}}}}&=F[P_{0},P_{1},P_{2},P_{3},\ldots ,P_{{\acute {n}}-3},P_{{\acute {n}}-2},P_{{\acute {n}}-1},P_{\acute {n}}],\\[10pt]&=F^{({\acute {n}})}(P_{0}<P<P_{\acute {n}})=\sum _{TN=1}^{UT=\infty }{\frac {F^{({\acute {n}})}(P_{(tn)})}{UT}}\\[10pt]&=F^{({\acute {n}})}(LB<P<UB)=G^{({\acute {n}}-1)}(LB<P<UB)=\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b375b60c7bae7c49c83567c022255d106be66acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:66.873ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {D^{\acute {n}}F(P_{0})}{DP^{\acute {n}}}}&=F[P_{0},P_{1},P_{2},P_{3},\ldots ,P_{{\acute {n}}-3},P_{{\acute {n}}-2},P_{{\acute {n}}-1},P_{\acute {n}}],\\[10pt]&=F^{({\acute {n}})}(P_{0}<P<P_{\acute {n}})=\sum _{TN=1}^{UT=\infty }{\frac {F^{({\acute {n}})}(P_{(tn)})}{UT}}\\[10pt]&=F^{({\acute {n}})}(LB<P<UB)=G^{({\acute {n}}-1)}(LB<P<UB)=\cdots \end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applying_the_divided_difference">Applying the divided difference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=10" title="Edit section: Applying the divided difference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{LB}^{UB}G(p)\,dp&=\int _{LB}^{UB}F'(p)\,dp=F(UB)-F(LB),\\[10pt]&=F[LB,UB]\Delta B,\\[10pt]&=F'(LB<P<UB)\Delta B,\\[10pt]&=\ G(LB<P<UB)\Delta B.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>B</mi> </mrow> </msubsup> <mi>G</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mi>B</mi> </mrow> </msubsup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>L</mi> <mi>B</mi> <mo>,</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">]</mo> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mi>G</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mi>B</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>U</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{LB}^{UB}G(p)\,dp&=\int _{LB}^{UB}F'(p)\,dp=F(UB)-F(LB),\\[10pt]&=F[LB,UB]\Delta B,\\[10pt]&=F'(LB<P<UB)\Delta B,\\[10pt]&=\ G(LB<P<UB)\Delta B.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997fd5c4d756e16ef9ffc40d18a4645961c4d0bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.493ex; margin-bottom: -0.178ex; width:51.44ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\int _{LB}^{UB}G(p)\,dp&=\int _{LB}^{UB}F'(p)\,dp=F(UB)-F(LB),\\[10pt]&=F[LB,UB]\Delta B,\\[10pt]&=F'(LB<P<UB)\Delta B,\\[10pt]&=\ G(LB<P<UB)\Delta B.\end{aligned}}}"></span></dd></dl> <p>Given that the mean value, derivative expression form provides all of the same information as the classical integral notation, the mean value form may be the preferable expression, such as in writing venues that only support/accept standard <a href="/wiki/ASCII" title="ASCII">ASCII</a> text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral). This is especially true for definite integrals that technically have (e.g.) 0 and either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049ec7aa24def176cbc6b6c6cb293f6a983ce88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.719ex; height:1.676ex;" alt="{\displaystyle \pi \,\!}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e525d75ed19439ca146a429294cd1724533812f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.882ex; height:2.176ex;" alt="{\displaystyle 2\pi \,\!}"></span> as boundaries, with the same divided difference found as that with boundaries of 0 and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\frac {\pi }{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {\pi }{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c27201a0e130111fa572eb5cc7e7b05789f6fef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.833ex; margin-bottom: -0.172ex; width:2.529ex; height:3.176ex;" alt="{\displaystyle {\begin{matrix}{\frac {\pi }{2}}\end{matrix}}}"></span> (thus requiring less averaging effort): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{0}^{2\pi }F'(p)\,dp&=4\int _{0}^{\frac {\pi }{2}}F'(p)\,dp=F(2\pi )-F(0)=4(F({\begin{matrix}{\frac {\pi }{2}}\end{matrix}})-F(0)),\\[10pt]&=2\pi F[0,2\pi ]=2\pi F'(0<P<2\pi ),\\[10pt]&=2\pi F[0,{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}]=2\pi F'(0<P<{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>F</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>P</mi> <mo><</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>F</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>P</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{0}^{2\pi }F'(p)\,dp&=4\int _{0}^{\frac {\pi }{2}}F'(p)\,dp=F(2\pi )-F(0)=4(F({\begin{matrix}{\frac {\pi }{2}}\end{matrix}})-F(0)),\\[10pt]&=2\pi F[0,2\pi ]=2\pi F'(0<P<2\pi ),\\[10pt]&=2\pi F[0,{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}]=2\pi F'(0<P<{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b8dfc785bfb525d1d7f054c11c4a579514d919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:68.478ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}\int _{0}^{2\pi }F'(p)\,dp&=4\int _{0}^{\frac {\pi }{2}}F'(p)\,dp=F(2\pi )-F(0)=4(F({\begin{matrix}{\frac {\pi }{2}}\end{matrix}})-F(0)),\\[10pt]&=2\pi F[0,2\pi ]=2\pi F'(0<P<2\pi ),\\[10pt]&=2\pi F[0,{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}]=2\pi F'(0<P<{\begin{matrix}{\frac {\pi }{2}}\end{matrix}}).\end{aligned}}}"></span></dd></dl> <p>This also becomes particularly useful when dealing with <i>iterated</i> and <a href="/wiki/Multiple_integral" title="Multiple integral"><i>multiple integral</i>s</a> (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{}\qquad \int _{CL}^{CU}\int _{BL}^{BU}\int _{AL}^{AU}F'(r,q,p)\,dp\,dq\,dr\\[10pt]&=\sum _{T\!C=1}^{U\!C=\infty }\left(\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }F^{'}(R_{(tc)}:Q_{(tb)}:P_{(ta)}){\frac {\Delta A}{U\!A}}\right){\frac {\Delta B}{U\!B}}\right){\frac {\Delta C}{U\!C}},\\[10pt]&=F'(C\!L<R<CU:BL<Q<BU:AL<P<\!AU)\Delta A\,\Delta B\,\Delta C.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mspace width="2em" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>U</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>U</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>U</mi> </mrow> </msubsup> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>C</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>:</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>:</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>A</mi> </mrow> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mrow> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>C</mi> </mrow> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>C</mi> <mspace width="negativethinmathspace" /> <mi>L</mi> <mo><</mo> <mi>R</mi> <mo><</mo> <mi>C</mi> <mi>U</mi> <mo>:</mo> <mi>B</mi> <mi>L</mi> <mo><</mo> <mi>Q</mi> <mo><</mo> <mi>B</mi> <mi>U</mi> <mo>:</mo> <mi>A</mi> <mi>L</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mspace width="negativethinmathspace" /> <mi>A</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>A</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>C</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{}\qquad \int _{CL}^{CU}\int _{BL}^{BU}\int _{AL}^{AU}F'(r,q,p)\,dp\,dq\,dr\\[10pt]&=\sum _{T\!C=1}^{U\!C=\infty }\left(\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }F^{'}(R_{(tc)}:Q_{(tb)}:P_{(ta)}){\frac {\Delta A}{U\!A}}\right){\frac {\Delta B}{U\!B}}\right){\frac {\Delta C}{U\!C}},\\[10pt]&=F'(C\!L<R<CU:BL<Q<BU:AL<P<\!AU)\Delta A\,\Delta B\,\Delta C.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276c5ed447c4e5489b878266550a21b6aabc6a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:68.343ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}&{}\qquad \int _{CL}^{CU}\int _{BL}^{BU}\int _{AL}^{AU}F'(r,q,p)\,dp\,dq\,dr\\[10pt]&=\sum _{T\!C=1}^{U\!C=\infty }\left(\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }F^{'}(R_{(tc)}:Q_{(tb)}:P_{(ta)}){\frac {\Delta A}{U\!A}}\right){\frac {\Delta B}{U\!B}}\right){\frac {\Delta C}{U\!C}},\\[10pt]&=F'(C\!L<R<CU:BL<Q<BU:AL<P<\!AU)\Delta A\,\Delta B\,\Delta C.\end{aligned}}}"></span></dd></dl> <p>Hence, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(R,Q:AL<P<AU)=\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R,Q:P_{(ta)})}{U\!A}};\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>Q</mi> <mo>:</mo> <mi>A</mi> <mi>L</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>A</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>Q</mi> <mo>:</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(R,Q:AL<P<AU)=\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R,Q:P_{(ta)})}{U\!A}};\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea8ec7f603976c13977962d569af2c3c7aad9d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.387ex; width:51.409ex; height:7.509ex;" alt="{\displaystyle F'(R,Q:AL<P<AU)=\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R,Q:P_{(ta)})}{U\!A}};\,\!}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(R:BL<Q<BU:AL<P<AU)=\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R:Q_{(tb)}:P_{(ta)})}{U\!A}}\right){\frac {1}{U\!B}}.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>R</mi> <mo>:</mo> <mi>B</mi> <mi>L</mi> <mo><</mo> <mi>Q</mi> <mo><</mo> <mi>B</mi> <mi>U</mi> <mo>:</mo> <mi>A</mi> <mi>L</mi> <mo><</mo> <mi>P</mi> <mo><</mo> <mi>A</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>R</mi> <mo>:</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>:</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>A</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>U</mi> <mspace width="negativethinmathspace" /> <mi>B</mi> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(R:BL<Q<BU:AL<P<AU)=\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R:Q_{(tb)}:P_{(ta)})}{U\!A}}\right){\frac {1}{U\!B}}.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2ac5d308b2f02c76bb971d18426e497770a75c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.387ex; width:82.61ex; height:7.509ex;" alt="{\displaystyle F'(R:BL<Q<BU:AL<P<AU)=\sum _{T\!B=1}^{U\!B=\infty }\left(\sum _{T\!A=1}^{U\!A=\infty }{\frac {F'(R:Q_{(tb)}:P_{(ta)})}{U\!A}}\right){\frac {1}{U\!B}}.\,\!}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Divided_differences" title="Divided differences">Divided differences</a></li> <li><a href="/wiki/Fermat_theory" class="mw-redirect" title="Fermat theory">Fermat theory</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Rectangle_method" class="mw-redirect" title="Rectangle method">Rectangle method</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient rule</a></li> <li><a href="/wiki/Symmetric_difference_quotient" class="mw-redirect" title="Symmetric difference quotient">Symmetric difference quotient</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-LaxTerrell2013-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-LaxTerrell2013_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFPeter_D._LaxMaria_Shea_Terrell2013" class="citation book cs1">Peter D. Lax; Maria Shea Terrell (2013). <i>Calculus With Applications</i>. Springer. p. 119. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-7946-8" title="Special:BookSources/978-1-4614-7946-8"><bdi>978-1-4614-7946-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+With+Applications&rft.pages=119&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-7946-8&rft.au=Peter+D.+Lax&rft.au=Maria+Shea+Terrell&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-HockettBock2005-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-HockettBock2005_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShirley_O._HockettDavid_Bock2005" class="citation book cs1">Shirley O. Hockett; David Bock (2005). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780764177668/page/44"><i>Barron's how to Prepare for the AP Calculus</i></a></span>. Barron's Educational Series. p. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780764177668/page/44">44</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7641-2382-5" title="Special:BookSources/978-0-7641-2382-5"><bdi>978-0-7641-2382-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Barron%27s+how+to+Prepare+for+the+AP+Calculus&rft.pages=44&rft.pub=Barron%27s+Educational+Series&rft.date=2005&rft.isbn=978-0-7641-2382-5&rft.au=Shirley+O.+Hockett&rft.au=David+Bock&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780764177668%2Fpage%2F44&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Ryan2010-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ryan2010_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Ryan2010" class="citation book cs1">Mark Ryan (2010). <i>Calculus Essentials For Dummies</i>. John Wiley & Sons. pp. 41–47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-64269-6" title="Special:BookSources/978-0-470-64269-6"><bdi>978-0-470-64269-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus+Essentials+For+Dummies&rft.pages=41-47&rft.pub=John+Wiley+%26+Sons&rft.date=2010&rft.isbn=978-0-470-64269-6&rft.au=Mark+Ryan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-NealGustafson2012-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-NealGustafson2012_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarla_NealR._GustafsonJeff_Hughes2012" class="citation book cs1">Karla Neal; R. Gustafson; Jeff Hughes (2012). <i>Precalculus</i>. Cengage Learning. p. 133. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-82662-0" title="Special:BookSources/978-0-495-82662-0"><bdi>978-0-495-82662-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus&rft.pages=133&rft.pub=Cengage+Learning&rft.date=2012&rft.isbn=978-0-495-82662-0&rft.au=Karla+Neal&rft.au=R.+Gustafson&rft.au=Jeff+Hughes&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Comenetz2002-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Comenetz2002_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Comenetz2002_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Comenetz2002_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_Comenetz2002" class="citation book cs1">Michael Comenetz (2002). <i>Calculus: The Elements</i>. World Scientific. pp. 71–76 and 151–161. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4904-5" title="Special:BookSources/978-981-02-4904-5"><bdi>978-981-02-4904-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+The+Elements&rft.pages=71-76+and+151-161&rft.pub=World+Scientific&rft.date=2002&rft.isbn=978-981-02-4904-5&rft.au=Michael+Comenetz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Pasch2010-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pasch2010_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoritz_Pasch2010" class="citation book cs1">Moritz Pasch (2010). <i>Essays on the Foundations of Mathematics by Moritz Pasch</i>. Springer. p. 157. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-9416-2" title="Special:BookSources/978-90-481-9416-2"><bdi>978-90-481-9416-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+on+the+Foundations+of+Mathematics+by+Moritz+Pasch&rft.pages=157&rft.pub=Springer&rft.date=2010&rft.isbn=978-90-481-9416-2&rft.au=Moritz+Pasch&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-WilsonAdamson2008-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-WilsonAdamson2008_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_C._WilsonScott_Adamson2008" class="citation book cs1">Frank C. Wilson; Scott Adamson (2008). <i>Applied Calculus</i>. Cengage Learning. p. 177. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-618-61104-1" title="Special:BookSources/978-0-618-61104-1"><bdi>978-0-618-61104-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Calculus&rft.pages=177&rft.pub=Cengage+Learning&rft.date=2008&rft.isbn=978-0-618-61104-1&rft.au=Frank+C.+Wilson&rft.au=Scott+Adamson&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-RubySellers2014-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-RubySellers2014_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-RubySellers2014_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTamara_Lefcourt_RubyJames_SellersLisa_KorfJeremy_Van_Horn2014" class="citation book cs1">Tamara Lefcourt Ruby; James Sellers; Lisa Korf; Jeremy Van Horn; Mike Munn (2014). <i>Kaplan AP Calculus AB & BC 2015</i>. Kaplan Publishing. p. 299. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61865-686-5" title="Special:BookSources/978-1-61865-686-5"><bdi>978-1-61865-686-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kaplan+AP+Calculus+AB+%26+BC+2015&rft.pages=299&rft.pub=Kaplan+Publishing&rft.date=2014&rft.isbn=978-1-61865-686-5&rft.au=Tamara+Lefcourt+Ruby&rft.au=James+Sellers&rft.au=Lisa+Korf&rft.au=Jeremy+Van+Horn&rft.au=Mike+Munn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-HungerfordShaw2008-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-HungerfordShaw2008_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HungerfordShaw2008_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_HungerfordDouglas_Shaw2008" class="citation book cs1">Thomas Hungerford; Douglas Shaw (2008). <i>Contemporary Precalculus: A Graphing Approach</i>. Cengage Learning. pp. 211–212. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-10833-7" title="Special:BookSources/978-0-495-10833-7"><bdi>978-0-495-10833-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Precalculus%3A+A+Graphing+Approach&rft.pages=211-212&rft.pub=Cengage+Learning&rft.date=2008&rft.isbn=978-0-495-10833-7&rft.au=Thomas+Hungerford&rft.au=Douglas+Shaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Krantz2014-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Krantz2014_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Krantz2014_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteven_G._Krantz2014" class="citation book cs1">Steven G. Krantz (2014). <i>Foundations of Analysis</i>. CRC Press. p. 127. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-2075-9" title="Special:BookSources/978-1-4822-2075-9"><bdi>978-1-4822-2075-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Analysis&rft.pages=127&rft.pub=CRC+Press&rft.date=2014&rft.isbn=978-1-4822-2075-9&rft.au=Steven+G.+Krantz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-GriewankWalther2008-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-GriewankWalther2008_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndreas_GriewankAndrea_Walther2008" class="citation book cs1">Andreas Griewank; <a href="/wiki/Andrea_Walther" title="Andrea Walther">Andrea Walther</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qMLUIsgCwvUC&pg=PA2"><i>Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition</i></a>. SIAM. pp. 2–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-659-7" title="Special:BookSources/978-0-89871-659-7"><bdi>978-0-89871-659-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Evaluating+Derivatives%3A+Principles+and+Techniques+of+Algorithmic+Differentiation%2C+Second+Edition&rft.pages=2-&rft.pub=SIAM&rft.date=2008&rft.isbn=978-0-89871-659-7&rft.au=Andreas+Griewank&rft.au=Andrea+Walther&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqMLUIsgCwvUC%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Lang1968-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lang1968_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerge_Lang1968" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a> (1968). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/analysisi0000lang"><i>Analysis 1</i></a></span>. Addison-Wesley Publishing Company. p. <a rel="nofollow" class="external text" href="https://archive.org/details/analysisi0000lang/page/56">56</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+1&rft.pages=56&rft.pub=Addison-Wesley+Publishing+Company&rft.date=1968&rft.au=Serge+Lang&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fanalysisi0000lang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-Hahn1994-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hahn1994_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrian_D._Hahn1994" class="citation book cs1">Brian D. Hahn (1994). <i>Fortran 90 for Scientists and Engineers</i>. Elsevier. p. 276. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-340-60034-4" title="Special:BookSources/978-0-340-60034-4"><bdi>978-0-340-60034-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fortran+90+for+Scientists+and+Engineers&rft.pages=276&rft.pub=Elsevier&rft.date=1994&rft.isbn=978-0-340-60034-4&rft.au=Brian+D.+Hahn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-ClaphamNicholson2009-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-ClaphamNicholson2009_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristopher_ClaphamJames_Nicholson2009" class="citation book cs1">Christopher Clapham; James Nicholson (2009). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/conciseoxforddic00clap"><i>The Concise Oxford Dictionary of Mathematics</i></a></span>. Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/conciseoxforddic00clap/page/n312">313</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-157976-9" title="Special:BookSources/978-0-19-157976-9"><bdi>978-0-19-157976-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Concise+Oxford+Dictionary+of+Mathematics&rft.pages=313&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-157976-9&rft.au=Christopher+Clapham&rft.au=James+Nicholson&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconciseoxforddic00clap&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADifference+quotient" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Donald C. Benson, <i>A Smoother Pebble: Mathematical Explorations</i>, Oxford University Press, 2003, p. 176.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Difference_quotient&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://cis.stvincent.edu/carlsond/ma109/diffquot.html">Saint Vincent College: Br. David Carlson, O.S.B.—<i>MA109 The Difference Quotient</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050912183919/http://cis.stvincent.edu/carlsond/ma109/diffquot.html">Archived</a> 2005-09-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://web.mat.bham.ac.uk/D.F.M.Hermans/msmxg6/ln/lnotes78.html">University of Birmingham: Dirk Hermans—<i>Divided Differences</i></a></li> <li>Mathworld: <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DividedDifference.html"><i>Divided Difference</i></a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Mean-ValueTheorem.html"><i>Mean-Value Theorem</i></a></li></ul></li> <li>University of Wisconsin: <a href="/wiki/Thomas_W._Reps" title="Thomas W. Reps">Thomas W. Reps</a> and Louis B. Rall — <a rel="nofollow" class="external text" href="http://www.cs.wisc.edu/wpis/abstracts/tr1415r.abs.html"><i>Computational Divided Differencing and Divided-Difference Arithmetics</i></a></li> <li><a rel="nofollow" class="external text" href="http://giraldi.org/derivata/derivata.html">Interactive simulator on difference quotient to explain the derivative</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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.navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style></div><div role="navigation" class="navbox" aria-labelledby="Sir_Isaac_Newton" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Isaac_Newton" title="Template:Isaac Newton"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Isaac_Newton" title="Template talk:Isaac Newton"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Isaac_Newton" title="Special:EditPage/Template:Isaac Newton"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sir_Isaac_Newton" style="font-size:114%;margin:0 4em"><a href="/wiki/Isaac_Newton" title="Isaac Newton">Sir Isaac Newton</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Publications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Fluxions</a></i> (1671)</li> <li><i><a href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De Motu</a></i> (1684)</li> <li><i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> (1687)</li> <li><i><a href="/wiki/Opticks" title="Opticks">Opticks</a></i> (1704)</li> <li><i><a href="/wiki/The_Queries" class="mw-redirect" title="The Queries">Queries</a></i> (1704)</li> <li><i><a href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica</a></i> (1707)</li> <li><i><a href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De Analysi</a></i> (1711)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Other writings</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones</a></i> (1661–1665)</li> <li>"<a href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">standing on the shoulders of giants</a>" (1675)</li> <li><i><a href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</a></i> (c. 1680)</li> <li>"<a href="/wiki/General_Scholium" title="General Scholium">General Scholium</a>" (1713; <i>"<a href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">hypotheses non fingo</a>"</i> )</li> <li><i><a href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">Ancient Kingdoms Amended</a></i> (1728)</li> <li><i><a href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">Corruptions of Scripture</a></i> (1754)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Contributions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a> <ul><li><a href="/wiki/Fluxion" title="Fluxion">fluxion</a></li></ul></li> <li><a href="/wiki/Impact_depth" title="Impact depth">Impact depth</a></li> <li><a href="/wiki/Inertia" title="Inertia">Inertia</a></li> <li><a href="/wiki/Newton_disc" title="Newton disc">Newton disc</a></li> <li><a href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</a> <ul><li><a href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a></li></ul></li> <li><a href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</a></li> <li><a href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</a></li> <li><a href="/wiki/Newton_scale" title="Newton scale">Newton scale</a></li> <li><a href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</a></li> <li><a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></li> <li><a href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</a></li> <li><a href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</a></li> <li><a href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</a></li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> <ul><li><a href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">post-Newtonian expansion</a></li> <li><a href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">parameterized</a></li> <li><a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a></li></ul></li> <li><a href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> <ul><li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws</a></li></ul></li> <li><a href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</a></li> <li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</a> <ul><li><a href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Apollonius's problem</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">truncated Newton method</a></li></ul></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</a></li> <li><a href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</a></li> <li><a href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</a></li> <li><a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</a></li> <li><a href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</a></li> <li><a href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</a></li> <li><a href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</a></li> <li><a href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</a></li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> <ul><li><a href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">generalized Gauss–Newton method</a></li></ul></li> <li><a href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</a></li> <li><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a></li> <li><a href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</a></li> <li><a href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</a></li> <li><a href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</a></li> <li><a href="/wiki/Power_number" title="Power number">Newton number</a> <ul><li><a href="/wiki/Kissing_number" title="Kissing number">kissing number problem</a></li></ul></li> <li><a class="mw-selflink selflink">Newton's quotient</a></li> <li><a href="/wiki/Parallelogram_of_force" title="Parallelogram of force">Parallelogram of force</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</a></li> <li><a href="/wiki/Absolute_space_and_time#Newton" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Luminiferous_aether" title="Luminiferous aether">Luminiferous aether</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Newtonian series</a> <ul><li><a href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">table</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Personal life</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</a> (birthplace)</li> <li><a href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</a> (home)</li> <li><a href="/wiki/Early_life_of_Isaac_Newton" title="Early life of Isaac Newton">Early life</a></li> <li><a href="/wiki/Later_life_of_Isaac_Newton" title="Later life of Isaac Newton">Later life</a></li> <li><a href="/wiki/Isaac_Newton%27s_apple_tree" title="Isaac Newton's apple tree">Apple tree</a></li> <li><a href="/wiki/Religious_views_of_Isaac_Newton" title="Religious views of Isaac Newton">Religious views</a></li> <li><a href="/wiki/Isaac_Newton%27s_occult_studies" title="Isaac Newton's occult studies">Occult studies</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Copernican_Revolution" title="Copernican Revolution">Copernican Revolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Relations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Catherine_Barton" title="Catherine Barton">Catherine Barton</a> (niece)</li> <li><a href="/wiki/John_Conduitt" title="John Conduitt">John Conduitt</a> (nephew-in-law)</li> <li><a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> (professor)</li> <li><a href="/wiki/William_Clarke_(apothecary)" title="William Clarke (apothecary)">William Clarke</a> (mentor)</li> <li><a href="/wiki/Benjamin_Pulleyn" title="Benjamin Pulleyn">Benjamin Pulleyn</a> (tutor)</li> <li><a href="/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a> (student)</li> <li><a href="/wiki/William_Whiston" title="William Whiston">William Whiston</a> (student)</li> <li><a href="/wiki/John_Keill" title="John Keill">John Keill</a> (disciple)</li> <li><a href="/wiki/William_Stukeley" title="William Stukeley">William Stukeley</a> (friend)</li> <li><a href="/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a> (friend)</li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> (friend)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">Depictions</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(Blake)" title="Newton (Blake)"><i>Newton</i> by Blake</a> (monotype)</li> <li><a href="/wiki/Newton_(Paolozzi)" title="Newton (Paolozzi)"><i>Newton</i> by Paolozzi</a> (sculpture)</li> <li><i><a href="/wiki/Isaac_Newton_Gargoyle" title="Isaac Newton Gargoyle">Isaac Newton Gargoyle</a></i></li> <li><i><a href="/wiki/Astronomers_Monument" title="Astronomers Monument">Astronomers Monument</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;"><a href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">Namesake</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</a></li> <li><a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a></li> <li><a href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</a></li> <li><a href="/wiki/Institute_of_Physics_Isaac_Newton_Medal" class="mw-redirect" title="Institute of Physics Isaac Newton Medal">Isaac Newton Medal</a></li> <li><a href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</a></li> <li><a href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</a></li> <li><a href="/wiki/XMM-Newton" title="XMM-Newton">XMM-Newton</a></li> <li><a href="/wiki/Sir_Isaac_Newton_Sixth_Form" title="Sir Isaac Newton Sixth Form">Sir Isaac Newton Sixth Form</a></li> <li><a href="/wiki/Statal_Institute_of_Higher_Education_Isaac_Newton" title="Statal Institute of Higher Education Isaac Newton">Statal Institute of Higher Education Isaac Newton</a></li> <li><a href="/wiki/Newton_International_Fellowship" title="Newton International Fellowship">Newton International Fellowship</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;vertical-align:top;">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><div class="div-col"> <div class="CategoryTreeTag" data-ct-options="{"mode":20,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Isaac_Newton" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Category:Isaac_Newton" 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