{"id":617,"date":"2025-10-09T15:25:26","date_gmt":"2025-10-09T15:25:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=617"},"modified":"2026-04-13T16:08:40","modified_gmt":"2026-04-13T16:08:40","slug":"the-chain-rule-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/the-chain-rule-apply-it\/","title":{"raw":"The Chain Rule: Apply It","rendered":"The Chain Rule: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the chain rule when you have one independent variable<\/li>\r\n \t<li>Apply the chain rule with multiple changing variables<\/li>\r\n \t<li>Find derivatives implicitly when working with functions of multiple variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Tracking Drone Flight Performance<\/h2>\r\nYou work as a technician for a company that operates delivery drones. Each drone's battery efficiency depends on multiple factors like altitude, speed, and air temperature\u2014all of which change over time during a flight. To optimize delivery routes and predict battery life, you need to understand how quickly efficiency changes as these variables shift. Additionally, the drone's flight path follows certain constraints that define relationships between position variables. Your task is to apply chain rule techniques to analyze how efficiency changes during flight and to find rates of change when variables are related implicitly.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">[ohm_question hide_question_numbers=1]322633[\/ohm_question]<\/p>\r\n&nbsp;\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322634[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322635[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322636[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322637[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the chain rule when you have one independent variable<\/li>\n<li>Apply the chain rule with multiple changing variables<\/li>\n<li>Find derivatives implicitly when working with functions of multiple variables<\/li>\n<\/ul>\n<\/section>\n<h2>Tracking Drone Flight Performance<\/h2>\n<p>You work as a technician for a company that operates delivery drones. Each drone&#8217;s battery efficiency depends on multiple factors like altitude, speed, and air temperature\u2014all of which change over time during a flight. To optimize delivery routes and predict battery life, you need to understand how quickly efficiency changes as these variables shift. Additionally, the drone&#8217;s flight path follows certain constraints that define relationships between position variables. Your task is to apply chain rule techniques to analyze how efficiency changes during flight and to find rates of change when variables are related implicitly.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">\n<p class=\"font-claude-response-body whitespace-normal break-words\"><iframe loading=\"lazy\" id=\"ohm322633\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322633&theme=lumen&iframe_resize_id=ohm322633&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322634\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322634&theme=lumen&iframe_resize_id=ohm322634&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322635&theme=lumen&iframe_resize_id=ohm322635&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322636&theme=lumen&iframe_resize_id=ohm322636&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322637\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322637&theme=lumen&iframe_resize_id=ohm322637&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-617","chapter","type-chapter","status-publish","hentry"],"part":428,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/617","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/617\/revisions"}],"predecessor-version":[{"id":2850,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/617\/revisions\/2850"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/428"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/617\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=617"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=617"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=617"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=617"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}