{"id":11836,"date":"2014-03-17T14:50:50","date_gmt":"2014-03-17T14:50:50","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11836"},"modified":"2021-12-26T16:34:21","modified_gmt":"2021-12-26T16:34:21","slug":"duas-regras-de-derivacao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11836","title":{"rendered":"Duas regras de deriva\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_11836' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11836' class='GTTabs_curr'><a  id=\"11836_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11836' ><a  id=\"11836_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11836'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Determine regras de deriva\u00e7\u00e3o que permitam calcular facilmente derivadas de fun\u00e7\u00f5es do tipo:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{f\\left( x \\right) = \\frac{k}{{x &#8211; a}}}&amp;{}&amp;{}&amp;{g\\left( x \\right) = \\frac{k}{{{x^2}}}}<br \/>\n\\end{array}\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11836' onClick='GTTabs_show(1,11836)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11836'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{f\\left( x \\right) = \\frac{k}{{x &#8211; a}}}&amp;{}&amp;{}&amp;{g\\left( x \\right) = \\frac{k}{{{x^2}}}}<br \/>\n\\end{array}\\]<\/p>\n<\/blockquote>\n<p>Seja $k$ constante e ${x_0} \\in \\mathbb{R}\\backslash \\left\\{ a \\right\\}$.<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{f&#8217;\\left( {{x_0}} \\right)}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{f\\left( {{x_0} + h} \\right) &#8211; f\\left( {{x_0}} \\right)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{k}{{{x_0} + h &#8211; a}} &#8211; \\frac{k}{{{x_0} &#8211; a}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{{k{x_0} &#8211; ka &#8211; k{x_0} &#8211; kh + ka}}{{\\left( {{x_0} + h &#8211; a} \\right)\\left( {{x_0} &#8211; a} \\right)}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{ &#8211; kh}}{{h\\left( {{x_0} + h &#8211; a} \\right)\\left( {{x_0} &#8211; a} \\right)}}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{ &#8211; k}}{{\\left( {{x_0} + h &#8211; a} \\right)\\left( {{x_0} &#8211; a} \\right)}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; k}}{{{{\\left( {{x_0} &#8211; a} \\right)}^2}}}}<br \/>\n\\end{array}\\]<\/p>\n<p>Logo:\\[\\begin{array}{*{20}{c}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{f:}&amp;{\\mathbb{R}\\backslash \\left\\{ a \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{k}{{x &#8211; a}}}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{f&#8217;:}&amp;{\\mathbb{R}\\backslash \\left\\{ a \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{ &#8211; k}}{{{{\\left( {x &#8211; a} \\right)}^2}}}}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<\/p>\n<p>Seja $k$ constante e ${x_0} \\in \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$.<\/p>\n<\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{g&#8217;\\left( {{x_0}} \\right)}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{g\\left( {{x_0} + h} \\right) &#8211; g\\left( {{x_0}} \\right)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{k}{{{{\\left( {{x_0} + h} \\right)}^2}}} &#8211; \\frac{k}{{{x_0}^2}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{{k{x_0}^2 &#8211; k{x_0}^2 &#8211; 2kh{x_0} &#8211; k{h^2}}}{{{x_0}^2{{\\left( {{x_0} + h} \\right)}^2}}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{ &#8211; 2kh{x_0} &#8211; k{h^2}}}{{h{x_0}^2{{\\left( {{x_0} + h} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{ &#8211; 2k{x_0} &#8211; kh}}{{{x_0}^2{{\\left( {{x_0} + h} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; 2k{x_0}}}{{{x_0}^4}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{ &#8211; 2k}}{{{x_0}^3}}}<br \/>\n\\end{array}\\]<\/p>\n<p>Logo:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{\\begin{array}{*{20}{l}}<br \/>\n{g:}&amp;{\\mathbb{R}\\backslash \\left\\{ 0 \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{k}{{{x^2}}}}<br \/>\n\\end{array}}&amp;{}&amp;{}&amp;{\\begin{array}{*{20}{l}}<br \/>\n{g&#8217;:}&amp;{\\mathbb{R}\\backslash \\left\\{ 0 \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to \\frac{{ &#8211; 2k}}{{{x^3}}}}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11836' onClick='GTTabs_show(0,11836)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine regras de deriva\u00e7\u00e3o que permitam calcular facilmente derivadas de fun\u00e7\u00f5es do tipo: \\[\\begin{array}{*{20}{c}} {f\\left( x \\right) = \\frac{k}{{x &#8211; a}}}&amp;{}&amp;{}&amp;{g\\left( x \\right) = \\frac{k}{{{x^2}}}} \\end{array}\\] Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19448,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,134],"tags":[422,145,368],"series":[],"class_list":["post-11836","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-11-o-ano","tag-derivadas-2","tag-regras-de-derivacao"],"views":1726,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/03\/Funcao-11-p73-ex2.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11836","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11836"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11836\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19448"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11836"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11836"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11836"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11836"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}