@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: a skos:ConceptScheme .
psr:-GXT3JJHT-8
  dc:modified "2023-08-02"^^xsd:date ;
  skos:broader psr:-BXH41LGM-B, psr:-K0PQKG10-G ;
  skos:definition """In calculus, the <b>inverse function rule</b> is a formula that expresses the derivative of the inverse of a bijective and differentiable function <span class="texhtml mvar" style="font-style:italic;">f</span> in terms of the derivative of <span class="texhtml mvar" style="font-style:italic;">f</span>. More precisely, if the inverse of <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> is denoted as <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^{-1}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="f^{-1}"></span>, where <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^{-1}(y)=x}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}(y)=x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b379e2f3dd9e116a5f051ebc1967b1039d93a81" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:3.176ex;" alt="{\\\\displaystyle f^{-1}(y)=x}"></span> if and only 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 f(x)=y}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>y</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f(x)=y}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5080a8b0a963407ea74ffa50702563771518d1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\\\\displaystyle f(x)=y}"></span>, then the inverse function rule is, in Lagrange's notation,
<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 \\\\left[f^{-1}\\
ight]'(a)={\\rac {1}{f'\\\\left(f^{-1}(a)\\
ight)}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mrow>
<br/>            <mo>[</mo>
<br/>            <msup>
<br/>              <mi>f</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mo>−<!-- − --></mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mo>]</mo>
<br/>          </mrow>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <msup>
<br/>                <mi>f</mi>
<br/>                <mo>′</mo>
<br/>              </msup>
<br/>              <mrow>
<br/>                <mo>(</mo>
<br/>                <mrow>
<br/>                  <msup>
<br/>                    <mi>f</mi>
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mo>−<!-- − --></mo>
<br/>                      <mn>1</mn>
<br/>                    </mrow>
<br/>                  </msup>
<br/>                  <mo stretchy="false">(</mo>
<br/>                  <mi>a</mi>
<br/>                  <mo stretchy="false">)</mo>
<br/>                </mrow>
<br/>                <mo>)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\left[f^{-1}\\
ight]'(a)={\\rac {1}{f'\\\\left(f^{-1}(a)\\
ight)}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662f1be67eb38c8d7ecf30fd8afda594671fbeec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:24.145ex; height:6.009ex;" alt="\\\\left[f^{{-1}}\\
ight]'(a)={\\rac  {1}{f'\\\\left(f^{{-1}}(a)\\
ight)}}"></span>.</dd></dl>
<br/>This formula holds in general whenever <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> is continuous and injective on an interval <span class="texhtml mvar" style="font-style:italic;">I</span>, with <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> being differentiable at <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^{-1}(a)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f^{-1}(a)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a868a75d3fce8eb888e4e3725c595c7ae0a444a0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.692ex; height:3.176ex;" alt="{\\\\displaystyle f^{-1}(a)}"></span>(<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 \\\\in I}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>I</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\in I}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115150f88cabd4b520dc0251c17fe061dc4d9897" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.367ex; height:2.176ex;" alt="{\\\\displaystyle \\\\in I}"></span>) and where<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'(f^{-1}(a))\\
eq 0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>≠<!-- ≠ --></mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f'(f^{-1}(a))\\
eq 0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189f0f7e3a5e06dd607e57666f484f2be9bfb7ac" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.768ex; height:3.176ex;" alt="{\\\\displaystyle f'(f^{-1}(a))\\
eq 0}"></span>. The same formula is also equivalent to the expression
<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 {\\\\mathcal {D}}\\\\left[f^{-1}\\
ight]={\\rac {1}{({\\\\mathcal {D}}f)\\\\circ \\\\left(f^{-1}\\
ight)}},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow>
<br/>          <mo>[</mo>
<br/>          <msup>
<br/>            <mi>f</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msup>
<br/>          <mo>]</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
<br/>                </mrow>
<br/>              </mrow>
<br/>              <mi>f</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>∘<!-- ∘ --></mo>
<br/>              <mrow>
<br/>                <mo>(</mo>
<br/>                <msup>
<br/>                  <mi>f</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mo>−<!-- − --></mo>
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mo>)</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathcal {D}}\\\\left[f^{-1}\\
ight]={\\rac {1}{({\\\\mathcal {D}}f)\\\\circ \\\\left(f^{-1}\\
ight)}},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b71cf74aa5480ff9d43a9823ceb5580914f923a5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:24.89ex; height:6.009ex;" alt="{\\\\displaystyle {\\\\mathcal {D}}\\\\left[f^{-1}\\
ight]={\\rac {1}{({\\\\mathcal {D}}f)\\\\circ \\\\left(f^{-1}\\
ight)}},}"></span></dd></dl>
<br/>where <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 {\\\\mathcal {D}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathcal {D}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\\\\mathcal {D}}"></span> denotes the unary derivative operator (on the space of functions) 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 \\\\circ }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo>∘<!-- ∘ --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\circ }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="\\\\circ "></span> denotes function composition. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Inverse_function_rule">https://en.wikipedia.org/wiki/Inverse_function_rule</a>)"""@en ;
  skos:inScheme psr: ;
  a skos:Concept ;
  dc:created "2023-08-02"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Inverse_function_rule> ;
  skos:prefLabel "règle de dérivation des fonctions réciproques"@fr, "inverse function rule"@en .

psr:-K0PQKG10-G
  skos:prefLabel "calcul différentiel"@fr, "differential calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-GXT3JJHT-8 .

psr:-BXH41LGM-B
  skos:prefLabel "fonction inverse"@fr, "reciprocal function"@en ;
  a skos:Concept ;
  skos:narrower psr:-GXT3JJHT-8 .