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<div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;">Dada la función definida implícitamente:<br /><br /></span></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"></span></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"></span></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"> </span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#f22«/mi»«mo»+«/mo»«mi»#g22«/mi»«mo»=«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span><span style="font-family: times new roman,times,serif;"> <br />Entonces, <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«/math»</span></span></span></span></span></span></span></span> </span></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>Obs.: Ingresa las funciones trigonométricas de la siguiente manera:</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>- seno(x) se ingresa sen(x)</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>-coseno(x) se ingresa cos(x)</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>-tangente(x) se ingresa tan(x)</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>-cosecante(x) se ingresa cosec(x)</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>-secante(x) se ingresa sec(x)</em></font></span></span></p> <p><span class="nolink"><span class="nolink"><font size="4" face="times new roman,times,serif" color="#990000"><em>-cotangente(x) se ingresa cot(x)</em></font></span></span></p></span></font></div>
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<p><font size="4" face="times new roman,times,serif"><i>Solución:</i></font></p> <div style="text-align: justify; font-style: italic;"><font size="4" face="times new roman,times,serif">Consideremos <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»+«/mo»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span>, si <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span> y <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span> son funciones derivables; entonces, <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span> también lo es y la derivada de <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span> está dada por:<br /><br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math»</span></span></span></span></span></span></span></span> (1)<br />Por otro lado, si <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»g«/mi»«mo»§nbsp;«/mo»«/math»</span></span></span></span></span></span></span></span>son dos funciones (derivables) en la variable "<span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»</span>"; entonces, la derivada del producto es:<br /><br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mfenced»«mrow»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»·«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»·«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»+«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»·«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math»</span></span></span></span></span></span></span></span> (2)<br /><br />En nuestro caso y utilizando la propiedad (1), la derivada de la función <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#f22«/mi»«mo»+«/mo»«mi»#g22«/mi»«mo»=«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span> está dada por:<br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#f22«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math»</span></span></span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#g22«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mn»0«/mn»«mo»)«/mo»«/math»</span></span></span></span></span></span></span></span> (3) (derivada de la función constante cero)<br />Determinemos entonces <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#f22«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span> y <span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#g22«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span> por separado. <br />Para ello, debemos tener en cuenta que la variable "<span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math»</span>" depende de la variable </font><font size="4" face="times new roman,times,serif"><font size="4">"<span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»</span>"</font> <br />y usando la propiedad (2) antes mencionada:<br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#f22«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#d11«/mi»«mo»+«/mo»«mi»#d12«/mi»«/math»</span></span></span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»·«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math»</span></span></span></span></span></span><br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#g22«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#d21«/mi»«mo»+«/mo»«mi»#d22«/mi»«/math»</span></span></span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»·«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math»</span></span></span></span></span></span><br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"></span></span></span></span></span></span><br /><br />Luego, según lo mencionado anteriormente y considerando la igualdad (3), obtenemos:</font></div> <div style="text-align: justify; font-style: italic;"><font size="4" face="times new roman,times,serif"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#d11«/mi»«mo»+«/mo»«mi»#d21«/mi»«mo»+«/mo»«mfenced»«mrow»«mi»#d12«/mi»«mo»+«/mo»«mi»#d22«/mi»«/mrow»«/mfenced»«/math»</span></span><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»·«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span></span></span></span></span></span></span></span></span><br /></font><font size="4" face="times new roman,times,serif"><i>Finalmente, despejando:<br /><br /><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink">«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»#d«/mi»«/math»</span></span></span></span></span></span></span></span> </i><br /><br /></font><font face="times new roman,times,serif"><br /></font></div>
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<div style="text-align: justify; font-style: italic; font-family: times new roman,times,serif;"><font size="4">Bien. Continúa de esa manera.</font></div>
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<div style="text-align: justify; font-style: italic; font-family: times new roman,times,serif;"><font size="4">Revisa el desarrollo para identificar tus errores. ¡Tú puedes!</font></div>
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