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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;"></span><span style="font-family: times new roman,times,serif;"><i><font face="times new roman,times,serif">Dada la función definida implícitamente:</font></i></span></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"><i><font face="times new roman,times,serif"> </font></i><br /></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¨»«mi»#f22«/mi»«mo»+«/mo»«mi»#g22«/mi»«mo»=«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;"> <br /><span class="nolink">Entonces, <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></font></div> <div style="text-align: justify;"><font size="4" style="font-style: italic;"><span style="font-family: times new roman,times,serif;"><span class="nolink"><span class="nolink"><span class="nolink"><span class="nolink"> <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></span></span></span></span></span></font></div>
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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;">Solución:<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;">Consideremos la función </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¨»«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 style="font-family: times new roman,times,serif;">. <br />Si </span><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 style="font-family: times new roman,times,serif;"> y </span><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 style="font-family: times new roman,times,serif;"> son funciones derivables, entonces:</span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">(1) </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»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">Por otra parte, si la función </span><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 style="font-family: times new roman,times,serif;"> se escribe como la compuesta de dos funciones, a saber </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¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mo»(«/mo»«mi»g«/mi»«mo»§#8728;«/mo»«mi»h«/mi»«mo»)«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;">, donde </span><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 style="font-family: times new roman,times,serif;"> y </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¨»«mi»h«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;"> son funciones derivables, entonces:</span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">(2) </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»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»g«/mi»«mo»§#8728;«/mo»«mi»h«/mi»«mo»)«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mfenced close=¨]¨ open=¨[¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«/mrow»«/mfenced»«mo»·«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math»</span></span></span>, </span></span></span><span style="font-family: times new roman,times,serif;">aquí la expresión </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¨»«mfenced close=¨]¨ open=¨[¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»g«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»)«/mo»«/mrow»«/mfenced»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;"> indica la derivada de </span><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»«/math»</span></span></span></span></span><span style="font-family: times new roman,times,serif;"> evaluada en </span><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 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;"> <br />Además, si <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»«/math»</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¨»«mi»g«/mi»«/math»</span></span></span></span></span></span> son dos funciones derivables, entonces el producto también es <i><font face="times new roman,times,serif">derivable y está dado por:<br /><br />(3) <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><br style="font-family: times new roman,times,serif;" /></font></i></span><font face="times new roman,times,serif"></font><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">Consideremos ahora nuestro caso para la función </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¨»«mi»#f22«/mi»«mo»+«/mo»«mi»#g22«/mi»«mo»=«/mo»«mn»0«/mn»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;">. <br /><br />La derivada de esta función utilizando la propiedad (1) </span><span style="font-family: times new roman,times,serif;">está dada por:</span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><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¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#f22«/mi»«mo»+«/mo»«mi»#g22«/mi»«mo»)«/mo»«mo»=«/mo»«mn»0«/mn»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§#8658;«/mo»«mo»§nbsp;«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#f22«/mi»«mo»)«/mo»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#g22«/mi»«mo»)«/mo»«mo»=«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«/mtd»«/mtr»«/mtable»«/math»</span></span></span></span></span></span><span style="font-family: times new roman,times,serif;"></span><br style="font-family: times new roman,times,serif;" /><span style="font-family: times new roman,times,serif;">Ahora debemos calcular la derivada de cada sumando anterior.<br /> Así:</span><br style="font-family: times new roman,times,serif;" /><br style="font-family: times new roman,times,serif;" /><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»#dfa«/mi»«/math»</span></span></span></span></span></span> <font face="times new roman,times,serif"><i><br />Para calcular la derivada del segundo sumando utilizamos las propiedades (2) y (3).<br /> Así:</i></font><span style="font-family: times new roman,times,serif;"></span><br style="font-family: times new roman,times,serif;" /><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»#g22«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#dfb«/mi»«mo»·«/mo»«mi»#ff2«/mi»«mo»+«/mo»«mi»#fb«/mi»«mo»·«/mo»«mi»#dff2«/mi»«/math»</span></span></span></span></span></span><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><br /><span style="font-family: times new roman,times,serif;">De lo anterior,<br /><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¨»«mi»#dfa«/mi»«mo»+«/mo»«mi»#dfb«/mi»«mo»·«/mo»«mi»#ff2«/mi»«mo»+«/mo»«mi»#fb«/mi»«mo»·«/mo»«mi»#dff2«/mi»«/math»</span></span></span></span></span></span><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 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»«mn»0«/mn»«/math»</span></span></span></span></span></span><br />Despejando, obtenemos:<br /><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»«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><br /><br /><br /></span></font><br /></div>
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<font size="4" face="times new roman,times,serif"><em>¡Excelente!</em></font>
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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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