tag:blogger.com,1999:blog-50904056443666852792024-02-19T02:22:25.030-08:00Derivadas en Calculo“Las matemáticas son una gimnasia del espíritu y una preparación para la filosofía”. Isócrates.Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-5090405644366685279.post-40320027618852272722011-07-25T21:31:00.001-07:002011-09-11T19:42:09.657-07:00Concepto y Practica de Derivadas<div class="MsoNormal"><span class="apple-style-span"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>Este es un video explicativo sobre el concepto de derivadas y algunos ejemplos para poner en práctica nuestro conocimiento.</b></span></span><br />
<span class="apple-style-span"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b><br />
</b></span></span><br />
<span class="apple-style-span"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>Enlace: </b></span></span><br />
<span class="apple-style-span"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><a href="http://vimeo.com/26941322">http://vimeo.com/26941322</a></span></span></div>Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.com0tag:blogger.com,1999:blog-5090405644366685279.post-32415732935303491562011-07-25T20:02:00.001-07:002011-09-11T15:42:01.674-07:00Como Derivar por Limites<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; mso-line-height-alt: 12.35pt;"><b><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: medium;"></span></b><br />
<div class="MsoNormal" style="line-height: 12.35pt; margin-bottom: .0001pt; margin-bottom: 0in;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: medium;"><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><b>Se empezara con un ejercicio no muy complejo, teniendo en cuenta la ecuación general de las derivadas<o:p></o:p></b></span></span></div><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b> </b></span><br />
<div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"> </span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;">f'(x)= lim <u>f(x+h)-f(x)</u><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"> </span><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;">h->0 h<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;">1. </span><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;">f(x)=<span style="letter-spacing: 0.7pt;"> </span><span style="letter-spacing: 0.7pt;">x^2 + 6x − 3</span><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;">Para poder derivar la </span><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;">ecuación, la x debe ser reemplazada por (x+h) y -f(x) es reemplazada por la función general.<o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><b>Entonces la ecuación nos quedara de la siguiente manera:<o:p></o:p></b></span></span></div><div class="MsoNormal" style="font-size: large; line-height: 12.35pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><b><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: 16px; line-height: 18px;"> </span><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"></span></b><br />
<div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> f'(x)= lim <u>(x+h)^2 -x^2</u>+<u>6(x+h) - 6x</u></span><span lang="ES-CO" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12pt;"><o:p></o:p></span></b></span></div><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt; line-height: 115%;"> h->0 h </span><br />
</b></span><br />
<div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><b> </b></span></span></div><b><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"></span></b><br />
<div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;">Enseguida se deberá operar la función ya que esta función se encuentra elevada al cuadrado, deberá desarrollarse por medio de bases de cuadrados perfectos, como no lo menciona el triangulo de pascal con cada una de las potencias.</span><span lang="ES-CO" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12pt;"><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><b>La ecuación nos quedara de la siguiente manera:<o:p></o:p></b></span></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;">f'(x)= lim <u>x^2+2xh+h^2-x^2</u>+<u>6x+6h - 6x</u></span><span style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12pt;"><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large; line-height: 13.7pt; margin-bottom: 0.0001pt;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> </span><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;">h->0 h</span><span lang="ES-CO" style="color: #333333; font-family: 'Times New Roman', serif; font-size: 12pt;"><o:p></o:p></span></b></span></div><div class="MsoNormal" style="font-size: large;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><b><br />
</b></span></div><div class="MsoNormal" style="font-size: large;"><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif;"><span lang="ES-CO" style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>La función nos queda de este modo debido al desarrollo de base de cuadrados perfectos donde el primer termino es elevado al cuadrado + 2 veces el primer termino por el segundo + el ultimo termino elevado al cuadrado.<o:p></o:p></b></span></span></div></div><b><br />
</b><br />
<b><br />
</b><br />
<div class="MsoNormal"><span lang="ES-CO" style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>Luego de este proceso eliminamos términos semejantes para que la ecuación nos quede del siguiente modo:<o:p></o:p></b></span></div><div class="MsoNormal"><span lang="ES-CO" style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b><br />
</b></span></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><b><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> </span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;">f'(x)= lim <u>2xh+h^2</u>+<u>6h</u></span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><o:p></o:p></span></b></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> h->0 h</span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><o:p></o:p></span></b></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><b><br />
</b></span></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"></span></b></div><div class="MsoNormal"><span lang="ES-CO" style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>En el momento de eliminar tanto la h del denominador como la del numerador se puede despejar el límite:<o:p></o:p></b></span></div><div class="MsoNormal"><span lang="ES-CO" style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b><br />
</b></span></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> f'(x)= lim 2x+0+6</span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><o:p></o:p></span></b></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"> h->0 </span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt;"><o:p></o:p></span></b></div><div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><b><br />
</b></span></div><b><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: 16px;">Para finalizar obtendremos la función derivada, dejando la ecuación como:</span><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"></span></b><br />
<div class="MsoNormal" style="line-height: 13.7pt; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><b><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"></span></b></span></div><div class="MsoNormal"><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><b><br />
</b></span></div><div class="MsoNormal"><span style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt;"><span class="apple-style-span"><span lang="ES-CO" style="color: #333333; font-family: Arial, sans-serif; font-size: 12pt; letter-spacing: 0.7pt; line-height: 115%;"><b>f'(x)= 2x+6</b></span></span></span></div><div style="color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 15px; line-height: 20px;"><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"></div></div>Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.com0tag:blogger.com,1999:blog-5090405644366685279.post-67342257947953389712011-07-24T11:23:00.001-07:002011-09-11T15:42:22.737-07:00Ejercicios para tu propio aprendizaje<div class="MsoNormal"><span class="apple-style-span"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>Calcular las derivadas de las siguientes funciones:</b></span></span></div><ol><li><span class="apple-style-span"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><b>y (x)=</b></span></span><span class="apple-converted-space"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"> </span></span><span class="apple-style-span"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">5x -2 </span></span></li>
<li><span class="apple-style-span"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span class="apple-style-span"><span style="font-family: Arial, sans-serif; font-size: 10pt; line-height: 14px;"><b><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;">f (x)</span></span><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;"> =</span></span></b></span></span></span><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span class="apple-style-span"><span style="font-family: Arial, sans-serif; font-size: 10pt; line-height: 14px;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;"> x³+2x-5</span></span></span></span></span></span></span></li>
<li><span class="apple-style-span"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span class="apple-style-span"><span style="font-family: Arial, sans-serif; font-size: 10pt; line-height: 14px;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;"><b>f (x) =</b> </span></span></span></span></span></span></span></span></span><span class="apple-style-span"><span style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span class="apple-style-span"><span style="font-family: Arial, sans-serif; font-size: 10pt; line-height: 14px;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: small; line-height: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 18px;">-2x+2</span></span></span></span></span></span></span></span></span></li>
<li><span class="Apple-style-span" style="font-family: Arial, sans-serif; font-size: 16px; line-height: 18px;"><b style="mso-bidi-font-weight: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">f (x) = </span></b><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">-2x²-5</span></span></li>
<li><span class="Apple-style-span" style="font-family: Arial, sans-serif; font-size: 16px; line-height: 18px;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><div class="MsoNormal"><b style="mso-bidi-font-weight: normal;"><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">y (x) = </span></b><span style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">x³+x²+x+1</span></div></span></span></li>
</ol><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgxBrkz5fRrvYa6u9VjTHN9FWcXfQ9kssvSQWtijftO6trDMscQuR8tA2WJbDqtZuX0Iw1h5ccYQll1dAXGr5gNR6BXDbstPhVuUNJlLsouywFHcfzlgQTInsTzQ-skwB2YA9a7qwc3Ehv/s1600/Derivative.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgxBrkz5fRrvYa6u9VjTHN9FWcXfQ9kssvSQWtijftO6trDMscQuR8tA2WJbDqtZuX0Iw1h5ccYQll1dAXGr5gNR6BXDbstPhVuUNJlLsouywFHcfzlgQTInsTzQ-skwB2YA9a7qwc3Ehv/s1600/Derivative.png" /></a></div><br />
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<div class="MsoNormal"><b style="font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;"><o:p></o:p></b></div><br />
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<div class="MsoNormal"><span style="font-family: Arial, sans-serif; font-size: 10pt; line-height: 115%;"><o:p></o:p></span></div>Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.com0tag:blogger.com,1999:blog-5090405644366685279.post-55032550529550219162011-07-23T11:27:00.000-07:002011-09-11T15:42:49.532-07:00Video Explicativo<span class="apple-style-span"><b><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif; font-size: 12pt; line-height: 115%;">Este es un vídeo que nos explica un poco más sobre el concepto de derivada</span>.</b></span><br />
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<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/yW-jtRgmrC8?feature=player_embedded' frameborder='0'></iframe></div>Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.com0tag:blogger.com,1999:blog-5090405644366685279.post-76129375338963051712011-07-23T10:57:00.000-07:002011-09-11T15:43:28.611-07:00Concepto de Derivada<div style="margin-bottom: .0001pt; margin: 0in;"><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif;"><b>La Derivada es la pendiente de la recta tangente de una función en un punto determinado.<o:p></o:p></b></span></div><div style="margin-bottom: .0001pt; margin: 0in;"><b><br />
</b></div><div style="margin-bottom: .0001pt; margin: 0in;"><b><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif;">Su Definición Matemática es la siguiente:</span><span lang="ES-CO" style="color: black; font-family: Arial, sans-serif;"><o:p></o:p></span></b></div><b><br />
</b><br />
<b><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: 16px; letter-spacing: 1px; line-height: 20px;"> f'(x)= lim </span><span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: 16px; letter-spacing: 1px; line-height: 20px;"><u>f(x+h)-f(x)</u></span></b><br />
<span class="Apple-style-span" style="color: #333333; font-family: Arial, sans-serif; font-size: 16px; letter-spacing: 1px; line-height: 20px;"><b> h->0 h</b></span><br />
<div class="MsoNormal"><br />
</div><br />
<br />
<div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh43o7_m3g9jnYN3Lyq-Dv5JlX8GEpWni1fuWZmXoG9rWw4x0uiTbqk30qjOYHbtDOmCupRlVkuzEVCRlPsw_VD5omDrv__USH79hRiiwjf7dEbPqu4szBIuxRKMpFr1QQ256mYdoUEmt67/s1600/derivada.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh43o7_m3g9jnYN3Lyq-Dv5JlX8GEpWni1fuWZmXoG9rWw4x0uiTbqk30qjOYHbtDOmCupRlVkuzEVCRlPsw_VD5omDrv__USH79hRiiwjf7dEbPqu4szBIuxRKMpFr1QQ256mYdoUEmt67/s320/derivada.jpg" width="320" /></a></div>Stepha Torreshttp://www.blogger.com/profile/03648218135291875612noreply@blogger.com0