{"id":594,"date":"2020-10-21T10:49:43","date_gmt":"2020-10-21T08:49:43","guid":{"rendered":"https:\/\/www.pealfa.duckdns.org\/wordpress\/?p=594"},"modified":"2020-10-21T10:49:43","modified_gmt":"2020-10-21T08:49:43","slug":"calculando-las-derivadas-de-nuevas-funciones","status":"publish","type":"post","link":"https:\/\/www.pealfa.duckdns.org\/wordpress\/?p=594","title":{"rendered":"Calculando las derivadas de nuevas funciones"},"content":{"rendered":"<style type=\"text\/css\">\nboton {\n border: none;\n background: rgba(0,0,0,0);\n color: #3a7999;\n box-shadow: inset 0 0 0 3px #3a7999;\n padding: 10px;\n font-size: 125%;\n border-radius: 5px;\n position: relative;\n box-sizing: border-box;\n}\n<\/style>\n<p><script type=\"text\/javascript\">function SINO(cual) {\n   var elElemento=document.getElementById(cual);\n   if(elElemento.style.display == 'block') {\n      elElemento.style.display = 'none';\n   } else {\n      elElemento.style.display = 'block';\n   }\n}\n<\/script><\/p>\n<p>Bueno, hoy un ratito para calcular la funci\u00f3n derivada. Vamos a ampliar nuestro cat\u00e1logo de f\u00f3rmulas y vamos a introducir la derivada de: la ra\u00edz cuadrada, la tangente y cotangente y, por \u00faltimo, las funciones trigonom\u00e9tricas rec\u00edprocas (los arcos).<\/p>\n<p>Concretamente:<\/p>\n<fieldset style=\"border: 1px solid #8A0808; padding: 10px;\">\n<p>Obtengamos la funci\u00f3n derivada de:<\/p>\n<ol>\n<li>\\(f\\left(x\\right) = \\sqrt{1+\\operatorname{sen}\\left(2x\\right)}\\)<\/li>\n<p><\/p>\n<li>\\(g\\left(x\\right) = 3\\tan\\left(2x\\right)\\)<\/li>\n<p><\/p>\n<li>\\(u\\left(x\\right) = 5\\operatorname{arcsen}\\left(x^2\\right)\\)<\/li>\n<p><\/p>\n<li>\\(v\\left(x\\right) = 3\\arctan\\left({\\rm e}^{-x}\\right)\\)<\/li>\n<\/ol>\n<\/fieldset>\n<p>Las tenemos en los siguientes dos v\u00eddeos. Evidentemente, muuuuuy despacito y repasando cositas, porque si conocemos la f\u00f3rmula de derivaci\u00f3n y dominamos la regla de la cadena est\u00e1n en un periquete:<\/p>\n<div style=\"text-align: center\">\n<table width=\"100%\">\n<tbody>\n<tr>\n<th width=\"50%\" style=\"border: none;\">Otras derivadas I<\/th>\n<th width=\"50%\" style=\"border: none;\">Otras derivadas II<\/th>\n<\/tr>\n<tr>\n<th width=\"50%\" style=\"border: none;\"><a title=\"Derivada de ra\u00edz y tangente\" href=\"https:\/\/youtu.be\/OPEIGleQMdo\" target=\"_blank\" rel=\"noopener noreferrer\"><img decoding=\"async\" src=\"https:\/\/i.ytimg.com\/vi\/OPEIGleQMdo\/hqdefault.jpg?sqp=-oaymwEZCPYBEIoBSFXyq4qpAwsIARUAAIhCGAFwAQ==&#038;rs=AOn4CLCACDpSP103gs_w1O9o9YgzvULI-A\" alt=\"V\u00eddeo de las derivadas de ra\u00edz y tangente\"><\/a><\/th>\n<th width=\"50%\" style=\"border: none;\"><a title=\"Derivadas de arco seno y arco tangente\" href=\"https:\/\/youtu.be\/lReXSy0-xe0\" target=\"_blank\" rel=\"noopener noreferrer\"><img decoding=\"async\" src=\"https:\/\/i.ytimg.com\/vi\/lReXSy0-xe0\/hqdefault.jpg?sqp=-oaymwEZCPYBEIoBSFXyq4qpAwsIARUAAIhCGAFwAQ==&#038;rs=AOn4CLAeiG3-lb9OZPL6tBitCBGZnvmIOw\" alt=\"V\u00eddeo de las derivadas de arco seno y arco tangente\"><\/a><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Para ver si has captado el procedimiento, intenta responder:<\/p>\n<form name=\"ejercicio\">\n<fieldset style=\"border: 1px solid #8A0808; padding: 10px;\">\n<div class=\"pregunta\">\n<div style=\"text-align: center;\">CUESTI\u00d3N<\/div>\n<p>Las derivadas de las funciones<br \/>\n\\[ f\\left(x\\right) = 2\\cot\\left(3x\\right) \\text{ , } g\\left(x\\right) = 7\\sec\\left(x^2\\right) \\]<br \/>\nson:<\/p>\n<table width=\"100%\" cellpadding=\"4\" cellspacing=\"0\">\n<tr>\n<td width=\"90%\" style=\"border: none;\">\n<p align=\"left\"><input name=\"p1\" onclick=\"document.ejercicio.feedback1.value=' &#10060; Incorrecto '\" type=\"radio\" \/> \\(f&#8217;\\left(x\\right) = -2\\tan\\left(3x\\right) \\text{ , } g&#8217;\\left(x\\right) = 7\\csc\\left(x^2\\right)\\)<\/p>\n<\/td>\n<td rowspan=\"2\" width=\"10%\" style=\"border: none;\">\n<p align=\"center\"><boton><a href=\"javascript:void(0);\" onclick=\"SINO('suger1')\" style=\"text-decoration: none;\" title=\"Ver sugerencia\">&#10067;<\/a><\/boton><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"90%\" style=\"border: none;\">\n<p align=\"left\"><input name=\"p1\" onclick=\"document.ejercicio.feedback1.value=' &#9989; Correcto '\" type=\"radio\" \/> \\(f&#8217;\\left(x\\right) = -6\\csc^2\\left(3x\\right) \\text{ , } g&#8217;\\left(x\\right) = 14x\\operatorname{sen}\\left(x^2\\right)\\sec^2\\left(x^2\\right)\\)<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"90%\" style=\"border: none;\">\n<p align=\"left\"><input name=\"p1\" onclick=\"document.ejercicio.feedback1.value=' &#10060; Incorrecto'\" type=\"radio\" \/> \\(f&#8217;\\left(x\\right) = -6\\tan\\left(3x\\right) \\text{ , } g&#8217;\\left(x\\right) = 14x\\csc\\left(x^2\\right)\\)<\/p>\n<\/td>\n<td rowspan=\"2\" width=\"10%\" style=\"border: none;\">\n<p align=\"center\"><boton><a href=\"javascript:void(0);\" onclick=\"SINO('solu1')\" style=\"text-decoration: none;\" title=\"Ver soluci\u00f3n\"> &#9997; <\/a><\/boton><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"90%\" style=\"border: none;\">\n<p align=\"left\"><input name=\"p1\" onclick=\"document.ejercicio.feedback1.value=' &#10060; Incorrecto '\" type=\"radio\" \/>\\(f&#8217;\\left(x\\right) = -2\\csc^2\\left(3x\\right) \\text{ , } g&#8217;\\left(x\\right) = 7\\operatorname{sen}\\left(x^2\\right)\\sec^2\\left(x^2\\right)\\)<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div style=\"text-align: center;\"><input name=\"feedback1\" size=\"10\" \/><\/div>\n<p>\n<\/div>\n<div id=\"suger1\" style=\"display: none;\">\n<hr \/>\n<p>Sugerencia:<\/p>\n<p>Primero obtenemos la derivada de la cotangente observando que es coseno entre seno, resultando la opuesta del cuadrado de la cosecante. Y, por otro lado, como la secante es la inversa del coseno, derivando el cociente obtenemos el producto del seno por el cuadrado de la secante.<\/p>\n<\/div>\n<div id=\"solu1\" style=\"display: none;\">\n<hr \/>\n<p>Soluci\u00f3n:<\/p>\n<p>Utilizando lo se\u00f1alado en la sugerencia anterior tenemos las derivadas de la cotangente y de la secante (est\u00e1n en cualquier tabla de derivadas). Aplicando la regla de la cadena:<\/p>\n<p>\\( f&#8217;\\left(x\\right) = 2\\cdot \\dfrac{-3}{\\operatorname{sen}^2\\left(3x\\right)}=-6\\csc^2\\left(3x\\right)  \\)<\/p>\n<p>\\( g&#8217;\\left(x\\right) = 7 \\cdot\\dfrac{2x\\operatorname{sen}\\left(x^2\\right)}{\\cos^2\\left(x^2\\right)}= 14x\\operatorname{sen}\\left(x^2\\right)\\sec^2\\left(x^2\\right)\\)<\/p>\n<\/div>\n<\/fieldset>\n<p>\n<\/form>\n<p>Bueno, espero que tanto los v\u00eddeos como la resoluci\u00f3n de este ejercicio haya sido provechosa.<\/p>\n<p>Gracias por la atenci\u00f3n y espero que nos veamos pronto.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bueno, hoy un ratito para calcular la funci\u00f3n derivada. Vamos a ampliar nuestro cat\u00e1logo de f\u00f3rmulas y vamos a introducir la derivada [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"post-templates\/post_nosidebar.php","format":"standard","meta":{"footnotes":""},"categories":[19],"tags":[57,58,50],"class_list":["post-594","post","type-post","status-publish","format-standard","hentry","category-matematicas-ii","tag-ampliacion-de-derivadas","tag-calculo-diferencial","tag-video"],"_links":{"self":[{"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/594","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=594"}],"version-history":[{"count":3,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/594\/revisions"}],"predecessor-version":[{"id":597,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/594\/revisions\/597"}],"wp:attachment":[{"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=594"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=594"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.pealfa.duckdns.org\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}