@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-V2R0LQSL-3
  skos:prefLabel "dérivée"@fr, "derivative"@en ;
  a skos:Concept ;
  skos:related psr:-NWWH35XW-D .

psr:-K0PQKG10-G
  skos:prefLabel "calcul différentiel"@fr, "differential calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-NWWH35XW-D .

psr:-NWWH35XW-D
  skos:inScheme psr: ;
  skos:related psr:-V2R0LQSL-3 ;
  skos:prefLabel "règle du produit"@fr, "product rule"@en ;
  skos:altLabel "règle de Leibniz"@fr, "Leibniz product rule"@en, "Leibniz rule"@en ;
  skos:definition """In calculus, the <b>product rule</b> (or <b>Leibniz rule</b> or <b>Leibniz product rule</b>) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (u\\\\cdot v)'=u'\\\\cdot v+u\\\\cdot v'}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi>v</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>u</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi>v</mi>
<br/>        <mo>+</mo>
<br/>        <mi>u</mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <msup>
<br/>          <mi>v</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (u\\\\cdot v)'=u'\\\\cdot v+u\\\\cdot v'}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af29b8270711bcbb0f0950b1fe5af1967b17cf5f" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:22.212ex; height:3.009ex;" alt="{\\\\displaystyle (u\\\\cdot v)'=u'\\\\cdot v+u\\\\cdot v'}"></div> or in Leibniz's notation as <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\rac {d}{dx}}(u\\\\cdot v)={\\rac {du}{dx}}\\\\cdot v+u\\\\cdot {\\rac {dv}{dx}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mi>d</mi>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi>v</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>u</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mi>v</mi>
<br/>        <mo>+</mo>
<br/>        <mi>u</mi>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>v</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {d}{dx}}(u\\\\cdot v)={\\rac {du}{dx}}\\\\cdot v+u\\\\cdot {\\rac {dv}{dx}}.}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808f4fede6066e704904310b27ee786192dba4c7" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -2.005ex; width:28.492ex; height:5.509ex;" alt="{\\\\displaystyle {\\rac {d}{dx}}(u\\\\cdot v)={\\rac {du}{dx}}\\\\cdot v+u\\\\cdot {\\rac {dv}{dx}}.}"></div>
<br/>The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Product_rule">https://en.wikipedia.org/wiki/Product_rule</a>)"""@en, """En analyse mathématique, la <b>règle du produit</b>, aussi appelée <b>règle de Leibniz</b>, est une formule utilisée afin de trouver les dérivées de produits de fonctions. Sous sa forme la plus simple, elle s'énonce ainsi&nbsp;:
<br/>
<br/><blockquote style="width:90%; border-left: solid #D0D0D0 1px; padding-left:1em;">
<br/>Soient <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> et <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 g}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>g</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle g}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="g"></span> deux fonctions réelles d'une variable réelle, dérivables en un point <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 x}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>. Alors leur produit <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 fg}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mi>g</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle fg}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="fg"></span> est aussi dérivable en <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 x}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> et <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 (fg)'(x)=f'(x)g(x)+f(x)g'(x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>f</mi>
<br/>        <mi>g</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mi>g</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (fg)'(x)=f'(x)g(x)+f(x)g'(x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bea0896500ca06a06fc1b53c412cdbf9bf399c6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:32.725ex; height:3.009ex;" alt="{\\\\displaystyle (fg)'(x)=f'(x)g(x)+f(x)g'(x)}"></span>.</blockquote>
<br/>En notation de Leibniz, cette formule s'écrit&nbsp;:
<br/>
<br/><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 {\\rac {\\\\mathrm {d} (f\\\\,g)}{\\\\mathrm {d} x}}(x)={\\rac {\\\\mathrm {d} f}{\\\\mathrm {d} x}}(x)g(x)+f(x){\\rac {\\\\mathrm {d} g}{\\\\mathrm {d} x}}(x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>f</mi>
<br/>              <mspace width="thinmathspace"></mspace>
<br/>              <mi>g</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>f</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>g</mi>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {\\\\mathrm {d} (f\\\\,g)}{\\\\mathrm {d} x}}(x)={\\rac {\\\\mathrm {d} f}{\\\\mathrm {d} x}}(x)g(x)+f(x){\\rac {\\\\mathrm {d} g}{\\\\mathrm {d} x}}(x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928fb3a8c629f74db2d9b257e0696392101ea134" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:37.665ex; height:5.843ex;" alt="{\\\\displaystyle {\\rac {\\\\mathrm {d} (f\\\\,g)}{\\\\mathrm {d} x}}(x)={\\rac {\\\\mathrm {d} f}{\\\\mathrm {d} x}}(x)g(x)+f(x){\\rac {\\\\mathrm {d} g}{\\\\mathrm {d} x}}(x)}"></span></dd></dl>
<br/>Une application importante de la règle du produit est la méthode d'intégration par parties. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/R%C3%A8gle_du_produit">https://fr.wikipedia.org/wiki/R%C3%A8gle_du_produit</a>)"""@fr ;
  skos:broader psr:-K0PQKG10-G ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/R%C3%A8gle_du_produit>, <https://en.wikipedia.org/wiki/Product_rule> ;
  a skos:Concept .