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Función derivada de las operaciones de funciones - Historial de revisiones
2024-03-29T10:46:45Z
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88.14.43.173: /* Derivada del cociente */
2008-04-09T19:00:41Z
<p><span class="autocomment">Derivada del cociente</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revisión de 19:00 9 abr 2008</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> <rigth></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left(</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left(</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 94:</td>
<td colspan="2" class="diff-lineno">Línea 94:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> </rigth></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td></tr>
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88.14.43.173
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72.252.60.185: /* Derivada del cociente */
2008-03-14T17:37:18Z
<p><span class="autocomment">Derivada del cociente</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revisión de 17:37 14 mar 2008</td>
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<tr><td colspan="2" class="diff-lineno">Línea 94:</td>
<td colspan="2" class="diff-lineno">Línea 94:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </<del style="color: red; font-weight: bold; text-decoration: none;">rigthr</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </<ins style="color: red; font-weight: bold; text-decoration: none;">rigth</ins>></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td></tr>
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72.252.60.185
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72.252.60.185: /* Derivada del cociente */
2008-03-14T17:37:00Z
<p><span class="autocomment">Derivada del cociente</span></p>
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<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 17:37 14 mar 2008</td>
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<tr><td colspan="2" class="diff-lineno">Línea 94:</td>
<td colspan="2" class="diff-lineno">Línea 94:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> ^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </<del style="color: red; font-weight: bold; text-decoration: none;">center</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </<ins style="color: red; font-weight: bold; text-decoration: none;">rigthr</ins>></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td></tr>
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72.252.60.185
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72.252.60.185: /* Derivada del cociente */
2008-03-14T17:36:26Z
<p><span class="autocomment">Derivada del cociente</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revisión de 17:36 14 mar 2008</td>
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<tr><td colspan="2" class="diff-lineno">Línea 87:</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> <<del style="color: red; font-weight: bold; text-decoration: none;">center</del>></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> <<ins style="color: red; font-weight: bold; text-decoration: none;">rigth</ins>></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left(</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \left(</div></td></tr>
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Fjmolina en 16:28 12 ene 2007
2007-01-12T16:28:42Z
<p></p>
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<td colspan='2' style="background-color: white; color:black;">Revisión de 16:28 12 ene 2007</td>
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<tr><td colspan="2" class="diff-lineno">Línea 85:</td>
<td colspan="2" class="diff-lineno">Línea 85:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; viene dada por la fórmula:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&nbsp; viene dada por la fórmula:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> </del><br/></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><br/></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> <center></div></td></tr>
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Fjmolina
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Fjmolina en 16:27 12 ene 2007
2007-01-12T16:27:33Z
<p></p>
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<td colspan='2' style="background-color: white; color:black;">← Revisión anterior</td>
<td colspan='2' style="background-color: white; color:black;">Revisión de 16:27 12 ene 2007</td>
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<tr><td colspan="2" class="diff-lineno">Línea 12:</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><center></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\left(</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\left(</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \, + \, \mathrm{g} \,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \, + \, \mathrm{g} \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\right)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>^\prime \, = \, \mathrm{f}^\prime \, + \, \mathrm{g}^\prime \,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>^\prime \, = \, \mathrm{f}^\prime \, + \, \mathrm{g}^\prime \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></center></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 37:</td>
<td colspan="2" class="diff-lineno">Línea 37:</td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\left(</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\left(</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \, - \, \mathrm{g} \,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \, - \, \mathrm{g} \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\right)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>^\prime \, = \, \mathrm{f}^\prime \, - \, \mathrm{g}^\prime \,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>^\prime \, = \, \mathrm{f}^\prime \, - \, \mathrm{g}^\prime \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>==Derivada del producto =</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>==Derivada del producto<ins style="color: red; font-weight: bold; text-decoration: none;">=</ins>=</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mathrm{g}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mathrm{g}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>, viene <del style="color: red; font-weight: bold; text-decoration: none;">dado </del>por la fórmula:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>, viene <ins style="color: red; font-weight: bold; text-decoration: none;">dada </ins>por la fórmula:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><center></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\left(</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\left(</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \cdot \mathrm{g} \,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \mathrm{f} \cdot \mathrm{g} \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\right)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>^\prime \, = \, \mathrm{f}^\prime \cdot \mathrm{g} \, + \, \mathrm{f} \cdot \mathrm{g}^\prime \,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>^\prime \, = \, \mathrm{f}^\prime \cdot \mathrm{g} \, + \, \mathrm{f} \cdot \mathrm{g}^\prime \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td colspan="2" class="diff-lineno">Línea 83:</td>
<td colspan="2" class="diff-lineno">Línea 83:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\frac{f}{g}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\frac{f}{g}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&nbsp; viene <del style="color: red; font-weight: bold; text-decoration: none;">dado </del>por la fórmula:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&nbsp; viene <ins style="color: red; font-weight: bold; text-decoration: none;">dada </ins>por la fórmula:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><center></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins><math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\left(</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\left(</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \frac{f}{g} \,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> \, \frac{f}{g} \,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\right)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>\right)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins>^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div></center></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> </ins></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Category:Matemáticas]]</div></td></tr>
</table>
Fjmolina
http://www.wikillerato.org/index.php?title=Funci%C3%B3n_derivada_de_las_operaciones_de_funciones&diff=3149&oldid=prev
Fjmolina en 16:25 12 ene 2007
2007-01-12T16:25:17Z
<p></p>
<a href="http://www.wikillerato.org/index.php?title=Funci%C3%B3n_derivada_de_las_operaciones_de_funciones&diff=3149&oldid=3148">(Diferencias entre revisiones)</a>
Fjmolina
http://www.wikillerato.org/index.php?title=Funci%C3%B3n_derivada_de_las_operaciones_de_funciones&diff=3148&oldid=prev
Fjmolina en 16:24 12 ene 2007
2007-01-12T16:24:48Z
<p></p>
<p><b>Página nueva</b></p><div><math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en el punto &nbsp;<br />
<math><br />
x \, = \, a<br />
</math><br />
, &nbsp; <br />
<math><br />
\mathrm{f}^\prime<br />
\left(<br />
\, a \,<br />
\right)<br />
</math><br />
, si existe, es el valor del limite:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\lim_{h \to 0} \frac{\mathrm{f}\left( \, a \, + \, h \, \right) \, - \,<br />
\mathrm{f} \left( \, a \, \right)}{h}<br />
</math>.<br />
</center><br />
<br />
<br/><br />
<br />
Si &nbsp;<br />
<math><br />
\mathrm{f}^\prime<br />
\left(<br />
\, a \,<br />
\right)<br />
</math><br />
&nbsp; es un número real, la función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; es derivable en &nbsp;<br />
<math><br />
x \, = \, a<br />
</math>.<br />
Si &nbsp;<br />
<math><br />
\mathrm{f}^\prime<br />
\left(<br />
\, a \,<br />
\right)<br />
</math><br />
&nbsp; no es un número real o el límite no existe, la función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; no es derivable en dicho punto.<br />
<br />
<br/><br />
<br />
==Ejemplo==<br />
<br />
<br/><br />
<br />
Calculemos la derivada de &nbsp;<br />
<math><br />
\mathrm{f}<br />
\left(<br />
\, x \,<br />
\right)<br />
\, = \, x^2 <br />
</math><br />
&nbsp; en &nbsp;<br />
<math><br />
x \, = \, 2<br />
</math>:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\mathrm{f}^\prime<br />
\left(<br />
\, 2 \,<br />
\right)<br />
\, = \, \lim_{h \to 0} \frac{\mathrm{f}\left( \, 2 \, + \, h \, \right) \, - \,<br />
\mathrm{f} \left( \, 2 \, \right)}{h} \, = \, \lim_{h \to 0} \frac<br />
{\left( \, 2 \, + \, h \, \right)^2 \, - \, 2^2}{h} \, = \, <br />
</math><br />
<br />
<br/><br />
<br />
<math><br />
\, = \, \lim_{h \to 0}<br />
\frac {\left( \, 4 \, + \, 4h \, + \, h^2 \, \right) \, - \, 4}{h} \, = \,<br />
\lim_{h \to 0} \frac {4h \, + \, h^2}{h} \, = \, \lim_{h \to 0}<br />
\left(<br />
\, h \, + 4 \, \,<br />
\right)<br />
\, = \, 4<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]<br />
%% }}}<br />
%% {{{ =tasas de variación<br />
<br />
==Tasa de variación media==<br />
<br />
<br/><br />
<br />
Supongamos que un coche de fórmula uno se mueve en una carretera totalmente recta. A<br />
distintas distancias de la salida se registran los tiempos de paso, obteniendose la<br />
siguiente tabla:<br />
<br />
<br/><br />
<br />
<center><br />
[[Imagen:tabla7.png]]<br />
</center><br />
<br />
<br/><br />
<br />
En este caso, la posición, &nbsp;<br />
<math><br />
y<br />
</math><br />
, se puede ver como una función, &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
, del tiempo, &nbsp;<br />
<math><br />
x<br />
</math>; es decir:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
y \, = \, \mathrm{f} \left( \, x \, \right)<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
La '''''tasa de variación media''''' de la posición en el intervalo de tiempo desde el<br />
instante &nbsp;<br />
<math><br />
9<br />
</math><br />
&nbsp; al instante &nbsp;<br />
<math><br />
13.4<br />
</math><br />
&nbsp; es:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\frac{\mathrm{f} \left( \, 13.4 \, \right) \, - \, \mathrm{f} \left( \, 9 \,<br />
\right)}{13.4 \, - \, 9} \, = \, \frac{6.7 \, - \, 4.5}{13.4 \, - \, 9} \, = \, 0.5<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
En general, la tasa de variación media de la función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en &nbsp;<br />
<math><br />
\left[<br />
\, a, \, b \,<br />
\right]<br />
</math><br />
&nbsp; se define como el cociente:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\frac{\mathrm{f} \left( \, b \, \right) \, - \, \mathrm{f} \left( \, a \,<br />
\right)}{b \, - \, a}<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
==Tasa de variación instantánea==<br />
<br />
<br/><br />
<br />
La '''''tasa de variación instantánea''''' de la función &nbsp;<br />
<math><br />
f<br />
</math><br />
&nbsp; en el punto &nbsp;<br />
<math><br />
x \, = \, a<br />
</math><br />
&nbsp; se obtiene haciendo tender &nbsp;<br />
<math><br />
b<br />
</math><br />
&nbsp; a &nbsp;<br />
<math><br />
a<br />
</math><br />
&nbsp; en la '''''tasa de variación media''''' de la función &nbsp;<br />
<math><br />
f<br />
</math><br />
&nbsp; en el intervalo &nbsp;<br />
<math><br />
\left[<br />
\, a, \, b \,<br />
\right]<br />
</math>; por tanto, la '''''tasa de variación instantánea''''' de la función &nbsp;<br />
<math><br />
f<br />
</math><br />
&nbsp; en el punto &nbsp;<br />
<math><br />
x \, = \, a<br />
</math><br />
&nbsp; es<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\lim_{h \to 0}\frac{\mathrm{f}\left( \, a \, + \, h \, \right) \, - \, \mathrm{f}\left( \, a \, \right)}{h}<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
que es precisamente la [[Definición de derivada|derivada]] de la función &nbsp;<br />
<math><br />
f<br />
</math><br />
&nbsp; en el punto &nbsp;<br />
<math><br />
x \, = \, a<br />
</math>.<br />
<br />
<br/><br />
<br />
NOTA: En el límite anterior &nbsp;<br />
<math><br />
b \, = \, a \, + \, h<br />
</math>.<br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]<br />
<br />
%% }}}<br />
%% {{{ =Derivadas de las funciones elementales<br />
<br />
<center><br />
[[Imagen:tablaDeDerivadas.png]]<br />
</center><br />
<br />
%% }}}<br />
%% {{{ =función derivada<br />
<br />
Si &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; es una función derivable en el intervalo &nbsp;<br />
<math><br />
\left(<br />
\, a, \, b \,<br />
\right)<br />
\subset R<br />
</math><br />
, la '''''función derivada''''' de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; es la que a cada &nbsp;<br />
<math><br />
x \in<br />
\left(<br />
\, a, \, b \,<br />
\right)<br />
</math><br />
&nbsp; le hace corresponder la [[Definición de derivada|derivada]] de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en dicho punto. Esta función se designa por &nbsp;<br />
<math><br />
\mathrm{f}^\prime \left( \, x \, \right)<br />
</math>.<br />
<br />
<br/><br />
<br />
Una función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; es '''''derivable''''' en el intervalo &nbsp;<br />
<math><br />
\left(<br />
\, a, \, b \,<br />
\right)<br />
</math><br />
&nbsp; si lo es en cada punto del intervalo.<br />
<br />
<br/><br />
<br />
Llamamos '''''derivada de segundo orden''''' de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; a la función derivada de &nbsp;<br />
<math><br />
\mathrm{f}^\prime <br />
</math>.<br />
Esta función se denota por &nbsp;<br />
<math><br />
\mathrm{f}^{\prime \prime}<br />
</math>.<br />
<br />
<br/><br />
<br />
<math><br />
\mathrm{f}^{\prime \prime \prime}<br />
</math><br />
&nbsp; es la '''''derivada tercera''''' de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; y, en general, &nbsp;<br />
<math><br />
\mathrm{f}^{\left( \, n \, \right)}<br />
</math><br />
&nbsp; es la '''''derivada n-ésima''''' de &nbsp;<br />
<math><br />
\mathrm{f} \left( \, x \, \right)<br />
</math>: &nbsp;<br />
<math><br />
\mathrm{f}^{\left( \, n \, \right)}<br />
</math><br />
&nbsp; es la función derivada de &nbsp;<br />
<math><br />
\mathrm{f}^{\left( \, n \, - \, 1 \, \right)}<br />
</math>.<br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]<br />
<br />
%% }}}<br />
%% {{{ =significado geométrico de la derivada<br />
<br />
Consideremos la grafica de una función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
. Tomemos un punto &nbsp;<br />
<math><br />
A \, = \,<br />
\left(<br />
\, A_x, \, \mathrm{f} \left( \, A_x \, \right) \,<br />
\right)<br />
</math><br />
&nbsp; en dicha grafica y consideremos una sucesión de puntos &nbsp;<br />
<math><br />
A_1, \, A_2, \, A_3, \, \ldots<br />
</math><br />
&nbsp; en la grafica de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
. Supongamos que todos estos puntos estan a la derecha de &nbsp;<br />
<math><br />
A<br />
</math><br />
&nbsp; y que cuando &nbsp;<br />
<math><br />
n \to \infty<br />
</math><br />
, &nbsp;<br />
<math><br />
A_n \to A <br />
</math>.<br />
<br />
<br/><br />
<br />
La recta que pasa por los puntos &nbsp;<br />
<math><br />
A<br />
</math><br />
&nbsp; y &nbsp;<br />
<math><br />
A_n<br />
</math><br />
&nbsp; es una secante a la grafica de la función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math>. De esta forma, hay una secante para cada punto &nbsp;<br />
<math><br />
A_n<br />
</math>. Sea &nbsp;<br />
<math><br />
s_n<br />
</math><br />
&nbsp; la recta que pasa por &nbsp;<br />
<math><br />
A<br />
</math><br />
&nbsp; y por &nbsp;<br />
<math><br />
A_n<br />
</math><br />
.<br />
<br />
<br/><br />
<br />
<center><br />
[[Imagen:tangente.png]]<br />
</center><br />
<br />
<br/><br />
<br />
Cuando &nbsp;<br />
<math><br />
n<br />
</math> <br />
&nbsp; tiende a &nbsp;<br />
<math><br />
\infty<br />
</math><br />
, &nbsp;<br />
<math><br />
s_n<br />
</math><br />
&nbsp; tiende a la tangente a la grafica de la función &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en el punto &nbsp;<br />
<math><br />
A<br />
</math>, &nbsp;<br />
<math><br />
t<br />
</math>:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
s_n \to t<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
Habria de esperar, pues, que la pendiente de &nbsp;<br />
<math><br />
s_n<br />
</math><br />
&nbsp; tienda a la pendiente de &nbsp;<br />
<math><br />
t<br />
</math><br />
&nbsp; cuando &nbsp;<br />
<math><br />
n<br />
</math> <br />
&nbsp; tiende a &nbsp;<br />
<math><br />
\infty<br />
</math>. Como la pendiente de &nbsp;<br />
<math><br />
s_n<br />
</math><br />
&nbsp; es una [[La derivada como una tasa de variación instantánea|tasa de variación<br />
media]]:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\frac{\mathrm{f} \left( \, A_{n,x} \, \right) \, - \, \mathrm{f} \left( \, A_x \,<br />
\right)}{A_{n,x} \, - \, A_x}<br />
</math><br />
<br />
<br/><br />
<br />
(<math><br />
A_{n,x} \, = <br />
</math><br />
&nbsp; abcisa de &nbsp;<br />
<math><br />
A_n<br />
</math>)<br />
</center><br />
<br />
<br/><br />
<br />
su limite cuando &nbsp;<br />
<math><br />
n \to \infty<br />
</math><br />
&nbsp; es una [[La derivada como una tasa de variación instantánea|tasa de variación instantánea]], la [[Definición de derivada|derivada]] de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en &nbsp;<br />
<math><br />
A_x<br />
</math>; es decir la pendiente de &nbsp;<br />
<math><br />
t<br />
</math><br />
&nbsp; es la derivada de &nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; en &nbsp;<br />
<math><br />
A_x<br />
</math>.<br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]<br />
<br />
%% }}}<br />
%% {{{ =función derivada de las operaciones de funciones<br />
<br />
__TOC__<br />
<br />
<br/><br />
<br />
==Derivada de la suma==<br />
<br />
<br/><br />
<br />
La [[Función derivada y derivadas sucesivas|derivada]] de la suma de dos funciones es<br />
igual a la suma de las derivadas de dichas funciones:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\left(<br />
\, \mathrm{f} \, + \, \mathrm{g} \,<br />
\right)<br />
^\prime \, = \, \mathrm{f}^\prime \, + \, \mathrm{g}^\prime \,<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
Este resultado, se puede ampliar a cualquier número de funciones.<br />
<br />
<br/><br />
<br />
==Derivada de la diferencia==<br />
<br />
<br/><br />
<br />
La [[Función derivada y derivadas sucesivas|derivada]] de la diferencia de dos funciones es<br />
igual a la diferencia de las derivadas de dichas funciones:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\left(<br />
\, \mathrm{f} \, - \, \mathrm{g} \,<br />
\right)<br />
^\prime \, = \, \mathrm{f}^\prime \, - \, \mathrm{g}^\prime \,<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
==Derivada del producto =<br />
<br />
<br/><br />
<br />
La [[Función derivada y derivadas sucesivas|derivada]] del producto de dos funciones,<br />
&nbsp;<br />
<math><br />
\mathrm{f}<br />
</math><br />
&nbsp; y &nbsp;<br />
<math><br />
\mathrm{g}<br />
</math><br />
, viene dado por la fórmula:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\left(<br />
\, \mathrm{f} \cdot \mathrm{g} \,<br />
\right)<br />
^\prime \, = \, \mathrm{f}^\prime \cdot \mathrm{g} \, + \, \mathrm{f} \cdot \mathrm{g}^\prime \,<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
==Derivada del cociente==<br />
<br />
<br/><br />
<br />
La [[Función derivada y derivadas sucesivas|derivada]] del cociente &nbsp;<br />
<math><br />
\frac{f}{g}<br />
</math><br />
&nbsp; viene dado por la fórmula:<br />
<br />
<br/><br />
<br />
<center><br />
<math><br />
\left(<br />
\, \frac{f}{g} \,<br />
\right)<br />
^\prime \, = \, \frac{\mathrm{f}^\prime \cdot \mathrm{g} \, - \, \mathrm{f} \cdot \mathrm{g}^\prime}{\mathrm{g}^2}<br />
</math><br />
</center><br />
<br />
<br/><br />
<br />
[[Category:Matemáticas]]</div>
Fjmolina