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class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Tangente hiperbólica</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 13 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-13" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">13 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9_%D8%A7%D9%84%D8%B8%D9%84_%D8%A7%D9%84%D8%B2%D8%A7%D8%A6%D8%AF%D9%8A%D8%A9" title="دالة الظل الزائدية (árabe)" lang="ar" hreflang="ar" data-title="دالة الظل الزائدية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hyperbolick%C3%BD_tangens" title="Hyperbolický tangens (checo)" lang="cs" hreflang="cs" data-title="Hyperbolický tangens" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Tangens_Hyperbolicus" title="Tangens Hyperbolicus (alemán)" lang="de" hreflang="de" data-title="Tangens Hyperbolicus" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70894304 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Hyperbolic_tangent" title="Hyperbolic tangent (inglés)" lang="en" hreflang="en" data-title="Hyperbolic tangent" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tangente_hiperboliko" title="Tangente hiperboliko (euskera)" lang="eu" hreflang="eu" data-title="Tangente hiperboliko" data-language-autonym="Euskara" data-language-local-name="euskera" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tangente_hyperbolique" title="Tangente hyperbolique (francés)" lang="fr" hreflang="fr" data-title="Tangente hyperbolique" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tanxente_hiperb%C3%B3lica" title="Tanxente hiperbólica (gallego)" lang="gl" hreflang="gl" data-title="Tanxente hiperbólica" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ja badge-Q70894304 mw-list-item" title=""><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A%E6%AD%A3%E6%8E%A5%E5%87%BD%E6%95%B0" title="双曲線正接函数 (japonés)" lang="ja" hreflang="ja" data-title="双曲線正接函数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9E%84%E1%9F%8B%E1%9E%9F%E1%9E%84%E1%9F%8B%E1%9E%A2%E1%9F%8A%E1%9E%B8%E1%9E%96%E1%9F%82%E1%9E%94%E1%9E%BC%E1%9E%9B%E1%9E%B8%E1%9E%80" title="តង់សង់អ៊ីពែបូលីក (jemer)" lang="km" hreflang="km" data-title="តង់សង់អ៊ីពែបូលីក" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="jemer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tangente_hiperb%C3%B3lica" title="Tangente hiperbólica (portugués)" lang="pt" hreflang="pt" data-title="Tangente hiperbólica" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tangens_hiperboli%C4%8Dni" title="Tangens hiperbolični (serbocroata)" lang="sh" hreflang="sh" data-title="Tangens hiperbolični" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BD%D0%B3%D0%B5%D0%BD%D1%81_%D1%85%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B8" title="Тангенс хиперболични 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a.image>img{max-width:100%!important;height:auto!important}</style><table class="infobox" style="width:22.7em; line-height: 1.4em; text-align:left; padding:.23em;"><tbody><tr><th colspan="3" class="cabecera física" style="text-align:center;background-color:transparent;color:inherit;">tangente hiperbólica</th></tr><tr><td colspan="3" class="imagen" style="text-align:center;"> <span typeof="mw:File"><a href="/wiki/Archivo:Hyperbolic_Tangent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Hyperbolic_Tangent.svg/250px-Hyperbolic_Tangent.svg.png" decoding="async" width="250" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Hyperbolic_Tangent.svg/375px-Hyperbolic_Tangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Hyperbolic_Tangent.svg/500px-Hyperbolic_Tangent.svg.png 2x" data-file-width="490" data-file-height="270" /></a></span><br /><div style="display:inline;"><a href="/wiki/Gr%C3%A1fica_de_una_funci%C3%B3n" title="Gráfica de una función">Gráfica</a> de tangente hiperbólica</div></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;">Definición</th><td colspan="2"> <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 \tanh x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204d72ada9b1537dde43543c3c29e5a08e8f7af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.756ex; height:2.176ex;" alt="{\displaystyle \tanh x\,}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Anexo:Funciones_matem%C3%A1ticas" title="Anexo:Funciones matemáticas">Tipo</a></th><td colspan="2"> <a href="/wiki/Funci%C3%B3n_real" title="Función real">Función real</a></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Dominio_de_definici%C3%B3n" class="mw-redirect" title="Dominio de definición">Dominio</a></th><td colspan="2"> <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 (-\infty ,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e577bfa9ed1c0f83ed643206abae3cd2f234cf9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.107ex; height:2.843ex;" alt="{\displaystyle (-\infty ,+\infty )}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Codominio" title="Codominio">Codominio</a></th><td colspan="2"> <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 (-1,1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc97c7af805d05e152a7f98ce9fc70ee8db0e7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.363ex; height:2.843ex;" alt="{\displaystyle (-1,1)\,}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Conjunto_imagen" class="mw-redirect" title="Conjunto imagen">Imagen</a></th><td colspan="2"> <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 (-1,1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc97c7af805d05e152a7f98ce9fc70ee8db0e7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.363ex; height:2.843ex;" alt="{\displaystyle (-1,1)\,}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;">Propiedades</th><td colspan="2"> <a href="/wiki/Funci%C3%B3n_biyectiva" title="Función biyectiva">Biyectiva</a> en el codominio<br /><a href="/wiki/Funci%C3%B3n_impar" class="mw-redirect" title="Función impar">Impar</a><br /><a href="/wiki/Funci%C3%B3n_mon%C3%B3tona" title="Función monótona">Estrictamente creciente</a><br /><a href="/wiki/Funci%C3%B3n_trascendente" title="Función trascendente">Trascendente</a></td></tr><tr><th colspan="3" style="text-align:center;background-color:#ccf;;"><a href="/wiki/C%C3%A1lculo_infinitesimal" title="Cálculo infinitesimal">Cálculo infinitesimal</a></th></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Derivada" title="Derivada">Derivada</a></th><td colspan="2"> <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 {1}{\cosh ^{2}x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\cosh ^{2}x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2279ddbd75b57f1f6e62ef975b981cb7ced7ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.011ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\cosh ^{2}x}}}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Integraci%C3%B3n_indefinida" title="Integración indefinida">Función primitiva</a></th><td colspan="2"> <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 \ln(\cosh x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(\cosh x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61dfcb85a1ac5913c0b40beee1ce771276b7193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.256ex; height:2.843ex;" alt="{\displaystyle \ln(\cosh x)\,}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/Funci%C3%B3n_inversa" title="Función inversa">Función inversa</a></th><td colspan="2"> <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 \operatorname {argtanh} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>argtanh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {argtanh} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1ee73247713c4cacdd4f68db7fbf75a9903fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.605ex; height:2.509ex;" alt="{\displaystyle \operatorname {argtanh} x}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;"><a href="/wiki/L%C3%ADmite_matem%C3%A1tico" class="mw-redirect" title="Límite matemático">Límites</a></th><td colspan="2"> <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 _{x\to -\infty }\tanh x=-1\,}"> <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>x</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }\tanh x=-1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3474109598497bd910edd52a4d0b9e510c2ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.717ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to -\infty }\tanh x=-1\,}"></span><br /><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 _{x\to +\infty }\tanh x=1\,}"> <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>x</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to +\infty }\tanh x=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35138448aa3b7c7e28620d320203f1238afa0dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.909ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to +\infty }\tanh x=1\,}"></span></td></tr><tr><th scope="row" style="text-align:left;background-color:#ddf;;">Funciones relacionadas</th><td colspan="2"> <a href="/wiki/Seno_hiperb%C3%B3lico" title="Seno hiperbólico">Seno hiperbólico</a><br /><a href="/wiki/Coseno_hiperb%C3%B3lico" title="Coseno hiperbólico">Coseno hiperbólico</a></td></tr><tr><td class="noprint" colspan="3" style="text-align:left;"><div class="plainlinks wikidata-link" style="font-size: 0.85em">[<a href="https://www.wikidata.org/wiki/Q1274703" class="extiw" title="d:Q1274703">editar datos en Wikidata</a>]</div></td></tr></tbody></table> <p>La <b>tangente hiperbólica</b> de un <a href="/wiki/N%C3%BAmero_real" title="Número real">número real</a> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> se designa mediante <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 \tanh x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0be1dd47ba059a32307fa7dbcf98c0877931e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.369ex; height:2.176ex;" alt="{\displaystyle \tanh x}"></span> y se define como el cociente entre el <a href="/wiki/Seno_hiperb%C3%B3lico" title="Seno hiperbólico">seno hiperbólico</a> y el <a href="/wiki/Coseno_hiperb%C3%B3lico" title="Coseno hiperbólico">coseno hiperbólico</a> del número real <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> </p><p>La fórmula es entonces </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 \tanh x={\cfrac {\sinh x}{\cosh x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>sinh</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>cosh</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x={\cfrac {\sinh x}{\cosh x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a038ab9784cee54e849c421b88f1dbd6199f7d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.424ex; height:7.176ex;" alt="{\displaystyle \tanh x={\cfrac {\sinh x}{\cosh x}}}"></span></dd></dl> <p>Si se sustituye de acuerdo con las definiciones de <a href="/wiki/Seno_hiperb%C3%B3lico" title="Seno hiperbólico">seno hiperbólico</a> y <a href="/wiki/Coseno_hiperb%C3%B3lico" title="Coseno hiperbólico">coseno hiperbólico</a>, se obtiene una fórmula más directa para la tangente hiperbólica, a saber: </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 \tanh x={\cfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x={\cfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7585a147635a0c2e0fb4268ef459f411e36a02d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.934ex; height:7.176ex;" alt="{\displaystyle \tanh x={\cfrac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Derivada">Derivada</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangente_hiperb%C3%B3lica&action=edit&section=1" title="Editar sección: Derivada"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><ul><li><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 {d}{dx}}\tanh x={\frac {1}{\cosh ^{2}x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\tanh x={\frac {1}{\cosh ^{2}x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a4b1bbcd35e8e2b762c7d3bb5644102f029c5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.247ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dx}}\tanh x={\frac {1}{\cosh ^{2}x}}}"></span></li></ul></dd></dl> <dl><dd><ul><li><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 {d}{dx}}\tanh x=1-\,\mathrm {tanh} ^{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\tanh x=1-\,\mathrm {tanh} ^{2}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf006301581d1477a56a7adc0667e20b1323cded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.662ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\tanh x=1-\,\mathrm {tanh} ^{2}x}"></span></li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Inversa">Inversa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangente_hiperb%C3%B3lica&action=edit&section=2" title="Editar sección: Inversa"><span>editar</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 {\mbox{arg tanh }}x=\ln \left({\sqrt {\frac {1+x}{1-x}}}\right)={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>arg tanh </mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{arg tanh }}x=\ln \left({\sqrt {\frac {1+x}{1-x}}}\right)={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff054c02b16ca5ea53b77a67611be5b95f45b2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.344ex; height:6.343ex;" alt="{\displaystyle {\mbox{arg tanh }}x=\ln \left({\sqrt {\frac {1+x}{1-x}}}\right)={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangente_hiperb%C3%B3lica&action=edit&section=3" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Trigonometr%C3%ADa" title="Trigonometría">Trigonometría</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangente_hiperb%C3%B3lica&action=edit&section=4" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Según el autor del libro puede escribirse como Tanh(x), tanh(x) o Tgh(x), también a veces se hace un espacio antes de la <b>h</b>, para lenguajes de programación se estila <b>tanh(x)</b> y para hojas Excel se estila <b>TANH(x)</b>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangente_hiperb%C3%B3lica&action=edit&section=5" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="Reference-Mathworld-Hyperbolic_Tangent" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HyperbolicTangent.html">«Hyperbolic Tangent»</a>. En Weisstein, Eric W, ed. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATangente+hiperb%C3%B3lica&rft.atitle=Hyperbolic+Tangent&rft.au=Weisstein%2C+Eric+W&rft.aulast=Weisstein%2C+Eric+W&rft.genre=article&rft.jtitle=MathWorld&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FHyperbolicTangent.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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