{"id":8657,"date":"2012-04-22T23:36:58","date_gmt":"2012-04-22T22:36:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8657"},"modified":"2021-12-30T10:15:00","modified_gmt":"2021-12-30T10:15:00","slug":"determine-as-expressoes-designatorias-das-funcoes-derivadas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8657","title":{"rendered":"Determine as express\u00f5es designat\u00f3rias das fun\u00e7\u00f5es derivadas"},"content":{"rendered":"<p><ul id='GTTabs_ul_8657' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8657' class='GTTabs_curr'><a  id=\"8657_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8657' ><a  id=\"8657_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_8657'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Determine as express\u00f5es designat\u00f3rias das fun\u00e7\u00f5es derivadas das fun\u00e7\u00f5es:\n<p>a) $f:x \\to \\operatorname{sen} (3x) + \\cos x$<\/p>\n<p>b) $g:x \\to {\\cos ^2}(2x)$<\/p>\n<p>c) $h:\\alpha\u00a0 \\to \\frac{{1 &#8211; \\cos (3\\alpha )}}{\\alpha }$<\/p>\n<p>d) $i:z \\to \\frac{{1 &#8211; \\cos (2z)}}{{1 + \\cos (2z)}}$<\/p>\n<p>e) $j:t \\to \\cos \\left( {4 &#8211; 3t} \\right)$<\/p>\n<\/li>\n<li>Sabendo que as fun\u00e7\u00f5es $f$ e $g$ s\u00e3o deriv\u00e1veis e que $g(1) = 3$, $g'(1) = 2$ e $f'(3) = 5$, determine, nos pontos indicados, o valor da derivada das fun\u00e7\u00f5es seguintes:\n<p>a) $f\\left( {g(x)} \\right)$ para $x = 1$;<\/p>\n<p>b) $f\\left( {g\\left( {2x + 1} \\right)} \\right)$ para $x = 0$.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8657' onClick='GTTabs_show(1,8657)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8657'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>a)<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{f'(x)}&amp; = &amp;{\\left( {\\operatorname{sen} (3x) + \\cos x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{\\cos (3x) \\times \\left( {3x} \\right)&#8217; &#8211; \\operatorname{sen} x} \\\\<br \/>\n{}&amp; = &amp;{3\\cos (3x) &#8211; \\operatorname{sen} x}<br \/>\n\\end{array}$$<br \/>\nb)<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{g'(x)}&amp; = &amp;{\\left( {{{\\cos }^2}(2x)} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{2 \\times \\cos (2x) \\times \\left( {\\cos (2x)} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{2\\cos (2x) \\times \\left( { &#8211; \\operatorname{sen} (2x) \\times \\left( {2x} \\right)&#8217;} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 2\\left( {2\\cos (2x) \\times \\operatorname{sen} (2x)} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 2\\operatorname{sen} (4x)}<br \/>\n\\end{array}$$<br \/>\nc)<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{h&#8217;\\left( \\alpha\u00a0 \\right)}&amp; = &amp;{{{\\left( {\\frac{{1 &#8211; \\cos (3\\alpha )}}{\\alpha }} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{sen} (3\\alpha ) \\times {{\\left( {3\\alpha } \\right)}^\\prime } \\times \\alpha\u00a0 &#8211; 1 \\times \\left( {1 &#8211; \\cos (3\\alpha )} \\right)}}{{{\\alpha ^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{3\\alpha \\operatorname{sen} (3\\alpha ) &#8211; 1 + \\cos (3\\alpha )}}{{{\\alpha ^2}}},}&amp;{\\alpha\u00a0 \\ne 0}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<br \/>\nd)<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{i&#8217;\\left( z \\right)}&amp; = &amp;{{{\\left( {\\frac{{1 &#8211; \\cos (2z)}}{{1 + \\cos (2z)}}} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{sen} (2z) \\times {{\\left( {2z} \\right)}^\\prime } \\times \\left( {1 + \\cos (2z)} \\right) + \\operatorname{sen} (2z) \\times {{\\left( {2z} \\right)}^\\prime } \\times \\left( {1 &#8211; \\cos (2z)} \\right)}}{{{{\\left( {1 + \\cos (2z)} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{2\\operatorname{sen} (2z) + 2\\operatorname{sen} (2z)\\cos (2z) + 2\\operatorname{sen} (2z) &#8211; 2\\operatorname{sen} (2z)\\cos (2z)}}{{{{\\left( {1 + \\cos (2z)} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{4\\operatorname{sen} (2z)}}{{{{\\left( {1 + \\cos (2z)} \\right)}^2}}},}&amp;{\\alpha\u00a0 \\ne \\frac{\\pi }{2} + k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}}<br \/>\n\\end{array}\\]<br \/>\ne)<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{j'(t)}&amp; = &amp;{\\left( {\\cos \\left( {4 &#8211; 3t} \\right)} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\operatorname{sen} \\left( {4 &#8211; 3t} \\right) \\times \\left( {4 &#8211; 3t} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{3\\operatorname{sen} \\left( {4 &#8211; 3t} \\right)}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>a)<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {f\\left( {g(x)} \\right)} \\right)}^\\prime }\\left( 1 \\right)}&amp; = &amp;{f&#8217;\\left( {g\\left( 1 \\right)} \\right) \\times g&#8217;\\left( 1 \\right)} \\\\<br \/>\n{}&amp; = &amp;{f&#8217;\\left( 3 \\right) \\times 2} \\\\<br \/>\n{}&amp; = &amp;{5 \\times 2} \\\\<br \/>\n{}&amp; = &amp;{10}<br \/>\n\\end{array}\\]<br \/>\nb)<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{{{\\left( {f\\left( {g\\left( {2x + 1} \\right)} \\right)} \\right)}^\\prime }\\left( 0 \\right)}&amp; = &amp;{f&#8217;\\left( {g\\left( {2 \\times 0 + 1} \\right)} \\right) \\times g&#8217;\\left( {2 \\times 0 + 1} \\right) \\times {{\\left( {2x + 1} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{f&#8217;\\left( {g\\left( 1 \\right)} \\right) \\times g&#8217;\\left( 1 \\right) \\times 2} \\\\<br \/>\n{}&amp; = &amp;{f&#8217;\\left( 3 \\right) \\times 2 \\times 2} \\\\<br \/>\n{}&amp; = &amp;{5 \\times 2 \\times 2} \\\\<br \/>\n{}&amp; = &amp;{20}<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<\/ol>\n<blockquote>\n<p>Se $f&#8217;$ e $g&#8217;$ existem e se $f \\circ g$ est\u00e1 definida, ent\u00e3o $$\\left( {f \\circ g} \\right)'(x) = f&#8217;\\left( {g(x)} \\right) \\times g'(x)$$<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8657' onClick='GTTabs_show(0,8657)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine as express\u00f5es designat\u00f3rias das fun\u00e7\u00f5es derivadas das fun\u00e7\u00f5es: a) $f:x \\to \\operatorname{sen} (3x) + \\cos x$ b) $g:x \\to {\\cos ^2}(2x)$ c) $h:\\alpha\u00a0 \\to \\frac{{1 &#8211; \\cos (3\\alpha )}}{\\alpha }$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19674,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,293,307],"series":[],"class_list":["post-8657","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-funcao-derivada","tag-funcoes-trigonometricas"],"views":4207,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat247.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8657","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=8657"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8657\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19674"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8657"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8657"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8657"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}