@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HLD5248R-P .

psr:-HLD5248R-P
  a skos:Concept ;
  dc:created "2023-08-01"^^xsd:date ;
  skos:related psr:-JRJF687X-3 ;
  skos:broader psr:-K0PQKG10-G, psr:-SKTRS1V0-R ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_d%C3%A9rivation_des_fonctions_compos%C3%A9es>, <https://en.wikipedia.org/wiki/Chain_rule> ;
  skos:altLabel "règle de dérivation en chaîne"@fr, "règle de la chaîne"@fr ;
  skos:inScheme psr: ;
  skos:definition """En mathématiques, dans le domaine de l'analyse, le théorème de dérivation des fonctions composées (parfois appelé règle de dérivation en chaîne ou règle de la chaîne, selon l'appellation anglaise) est une formule explicitant la dérivée d'une fonction composée pour deux fonctions dérivables. Elle permet de connaître la <i>j</i>-ème dérivée partielle de la <i>i</i>-ème application partielle de la composée de deux fonctions de plusieurs variables chacune. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_d%C3%A9rivation_des_fonctions_compos%C3%A9es">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_d%C3%A9rivation_des_fonctions_compos%C3%A9es</a>)"""@fr, """In calculus, the <b>chain rule</b> is a formula that expresses the derivative of the composition of two differentiable functions <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> in terms of the derivatives of <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span>. More precisely, if <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 h=f\\\\circ g}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>h</mi>
         <mo>=</mo>
         <mi>f</mi>
         <mo>∘<!-- ∘ --></mo>
         <mi>g</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle h=f\\\\circ g}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14345f3938c66b1fde08e18d8b301a59c5775a5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.027ex; height:2.509ex;" alt="{\\\\displaystyle h=f\\\\circ g}"></span> is the function such that <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 h(x)=f(g(x))}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>h</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>=</mo>
         <mi>f</mi>
         <mo stretchy="false">(</mo>
         <mi>g</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle h(x)=f(g(x))}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ac4cb5a6518f4fbf566c6b0d2f0af5d6551f0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.919ex; height:2.843ex;" alt="{\\\\displaystyle h(x)=f(g(x))}"></span> for every <span class="texhtml mvar" style="font-style:italic;">x</span>, then the chain rule is, in Lagrange's notation,
         </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 h'(x)=f'(g(x))g'(x).}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>h</mi>
         <mo>′</mo>
         </msup>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>=</mo>
         <msup>
         <mi>f</mi>
         <mo>′</mo>
         </msup>
         <mo stretchy="false">(</mo>
         <mi>g</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo stretchy="false">)</mo>
         <msup>
         <mi>g</mi>
         <mo>′</mo>
         </msup>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>.</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle h'(x)=f'(g(x))g'(x).}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c055a07905ad198e6ba0022fc32736cfd23daf4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.919ex; height:3.009ex;" alt="{\\\\displaystyle h'(x)=f'(g(x))g'(x).}"></span></dd></dl>
         <p>or, equivalently,
         </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 h'=(f\\\\circ g)'=(f'\\\\circ g)\\\\cdot g'.}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msup>
         <mi>h</mi>
         <mo>′</mo>
         </msup>
         <mo>=</mo>
         <mo stretchy="false">(</mo>
         <mi>f</mi>
         <mo>∘<!-- ∘ --></mo>
         <mi>g</mi>
         <msup>
         <mo stretchy="false">)</mo>
         <mo>′</mo>
         </msup>
         <mo>=</mo>
         <mo stretchy="false">(</mo>
         <msup>
         <mi>f</mi>
         <mo>′</mo>
         </msup>
         <mo>∘<!-- ∘ --></mo>
         <mi>g</mi>
         <mo stretchy="false">)</mo>
         <mo>⋅<!-- ⋅ --></mo>
         <msup>
         <mi>g</mi>
         <mo>′</mo>
         </msup>
         <mo>.</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle h'=(f\\\\circ g)'=(f'\\\\circ g)\\\\cdot g'.}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767d917fb4f1e4bb20a60cebc2df701874d44e3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.558ex; height:3.009ex;" alt="{\\\\displaystyle h'=(f\\\\circ g)'=(f'\\\\circ g)\\\\cdot g'.}"></span></dd></dl>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Chain_rule">https://en.wikipedia.org/wiki/Chain_rule</a>)"""@en ;
  dc:modified "2023-08-01"^^xsd:date ;
  skos:prefLabel "chain rule"@en, "théorème de dérivation des fonctions composées"@fr .

psr:-K0PQKG10-G
  skos:prefLabel "calcul différentiel"@fr, "differential calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-HLD5248R-P .

psr: a skos:ConceptScheme .
psr:-JRJF687X-3
  skos:prefLabel "formule de Faà di Bruno"@fr, "Faà di Bruno's formula"@en ;
  a skos:Concept ;
  skos:related psr:-HLD5248R-P .