{"id":2422,"date":"2023-04-20T09:59:45","date_gmt":"2023-04-20T07:59:45","guid":{"rendered":"https:\/\/mxth.dk\/?p=2422"},"modified":"2023-04-24T13:42:05","modified_gmt":"2023-04-24T11:42:05","slug":"andre-generelle-regneregler","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=2422","title":{"rendered":"Andre generelle regneregler"},"content":{"rendered":"\n<p class=\" eplus-wrapper\">Vi har tidligere set p\u00e5 de element\u00e6rer regneregler som g\u00e6lder for specifikke funktionstyper og p\u00e5 enkelte generelle regneregler, mere specifikt regnereglen for multiplikation med en konstant $\\bigl( f(x)=k\\cdot g(x)\\bigr)$ og summe eller differens $\\bigl(f(x)=g(x)\\pm h(x)\\bigr)$ og g\u00e6lder for alle funktionstyper.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Vi vil her udvide vores bibliotek af generelle regneregler til at omfatte yderligere tre, nemlig for produktet af funktioner (<em>produktreglen<\/em>), for kvortient af funktioner (kvo<em>tientreglen<\/em>) og for sammensatte funktioner (<em>k\u00e6dereglen<\/em>).<\/p>\n\n\n\n<h5 class=\" wp-block-heading eplus-wrapper\">Produktreglen<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">For funktioner som er ganget med hinanden, det vil sige er p\u00e5 formen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f(x)=g(x)\\cdot h(x)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">kan vi finde den afledte med f\u00f8lgende regneregl<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f'(x)=g'(x)\\cdot h(x) + g(x)\\cdot h'(x)$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=2430#DRPR001\" taget=\"_blank\" rel=\"noopener noreferrer\">DRPR001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c545\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.12<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c2723\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.13<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c2859\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.14<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c536\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.15<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\" wp-block-heading eplus-wrapper\">Kvotientreglen<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">For funktioner som er divideret med hinanden, det vil sige er p\u00e5 formen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f(x)=\\frac{g(x)}{h(x)}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">kan vi finde den afledte med f\u00f8lgende regneregl<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f'(x)=\\frac{g'(x)\\cdot h(x)-g(x)\\cdot h'(x)}{(g(x))^2}$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=1465#c12698\" taget=\"_blank\" rel=\"noopener noreferrer\">matBhtx 9.14<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=1465#c12704\" taget=\"_blank\" rel=\"noopener noreferrer\">matBhtx 9.20<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=132#c579\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.40<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\" wp-block-heading eplus-wrapper\">K\u00e6dereglen<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">For sammensatte funktioner, det vil sige funktioner hvor en funktion st\u00e5r inden i en anden funktion og derfor er p\u00e5 formen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f(x)=\\bigl(g\\circ h)(x)=g(h(x)\\bigr)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">kan vi benytte regnereglen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$f'(x)=g&#8217;\\bigl(h(x)\\bigr)\\cdot h'(x)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Da vi ogs\u00e5 har en anden notation for den afledte som er $f'(x)=\\frac{dy}{dx}$ kan vi ogs\u00e5 udtrykke k\u00e6dereglen for f\u00f8lgende form<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">$\\frac{dy}{dx}=\\frac{dy}{du}\\cdot \\frac{du}{dx}$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=1465#c14155\" taget=\"_blank\" rel=\"noopener noreferrer\">matBhtx 9.8<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=2430#DRKR001\" taget=\"_blank\" rel=\"noopener noreferrer\">DRKR001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=2430#DRKR002\" taget=\"_blank\" rel=\"noopener noreferrer\">DRKR002<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h3 class=\" wp-block-heading eplus-wrapper\">Blandede opgaver<\/h3>\n\n\n\n<p class=\" eplus-wrapper\">Her til sidst er der en r\u00e6kke blandede opgaver, hvor I selv skal afg\u00f8re hvilken regneregl I skal bruge.<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=1465#c12701\" taget=\"_blank\" rel=\"noopener noreferrer\">matBhtx 9.17<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c547\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.17<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener noreferrer\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=131#c550\" taget=\"_blank\" rel=\"noopener noreferrer\">matAB2stx 3.18<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=2430#DRBOR001\" taget=\"_blank\" rel=\"noopener noreferrer\">DRBOR001<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n","protected":false},"excerpt":{"rendered":"<p>Vi vil her udvide vores bibliotek af generelle regneregler til at omfatte yderligere tre, nemlig for produktet af funktioner (produktreglen), for kvortient af funktioner (kvotientreglen) og for sammensatte funktioner (k\u00e6dereglen).<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[11,3],"tags":[],"class_list":["post-2422","post","type-post","status-publish","format-standard","hentry","category-differentialregning","category-matematik"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2422","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2422"}],"version-history":[{"count":7,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2422\/revisions"}],"predecessor-version":[{"id":2438,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2422\/revisions\/2438"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2422"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2422"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}