145 The Chain Rule

>> YOUR LINK HERE: ___ http://youtube.com/watch?v=k0N3cPAp54I

Objectives: • 9. Use the Chain Rule to find partial derivatives. • 10. Find partial derivatives for implicitly-defined equations.","styleRuns":[{"startIndex":0,"length":125,"styleRunExtensions":{"styleRunColorMapExtension":{"colorMap":[{"key":"USER_INTERFACE_THEME_DARK","value":4294967295},{"key":"USER_INTERFACE_THEME_LIGHT","value":4279440147}]}},"fontFamilyName":"Roboto"}]},"headerRuns":[{"startIndex":0,"length":125,"headerMapping":"ATTRIBUTED_STRING_HEADER_MAPPING_UNSPECIFIED"}]}},{"itemSectionRenderer":{"contents":[{"continuationItemRenderer":{"trigger":"CONTINUATION_TRIGGER_ON_ITEM_SHOWN","continuationEndpoint":{"clickTrackingParams":"COMBELsvGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/next"}},"continuationCommand":{"token":"Eg0SC2swTjNjUEFwNTRJGAYyJSIRIgtrME4zY1BBcDU0STAAeAJCEGNvbW1lbnRzLXNlY3Rpb24%3D","request":"CONTINUATION_REQUEST_TYPE_WATCH_NEXT"}}}}],"trackingParams":"COMBELsvGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","sectionIdentifier":"comment-item-section","targetId":"comments-section"}}],"trackingParams":"COIBELovIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},"secondaryResults":{"secondaryResults":{"results":[{"compactVideoRenderer":{"videoId":"BF1d6A18AFM","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/BF1d6A18AFM/hqdefault.jpg?sqp=-oaymwEbCKgBEF5IVfKriqkDDggBFQAAiEIYAXABwAEG rs=AOn4CLBJ95FlPApRvJSwIh7cDvA4kjYatQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/BF1d6A18AFM/hqdefault.jpg?sqp=-oaymwEcCNACELwBSFXyq4qpAw4IARUAAIhCGAFwAcABBg== rs=AOn4CLC_7iNq1CwTrYv4VLZrVh2wt0Vxzg","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"14.6: Directional Derivatives and the Gradient Vector (1/2) by Alexandra Niedden 46,065 views 5 years ago 42 minutes"}},"simpleText":"14.6: Directional Derivatives and the Gradient Vector (1/2)"},"longBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"publishedTimeText":{"simpleText":"5 years ago"},"viewCountText":{"simpleText":"46,065 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"42 minutes, 15 seconds"}},"simpleText":"42:15"},"navigationEndpoint":{"clickTrackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWRIgs-ngY_u3aGTAZoBBQgBEPgd","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=BF1d6A18AFM","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"BF1d6A18AFM","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 id=045d5de80d7c0053 ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1643750 mt=1741281175 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_n4nO9hINEkA9rVfShhbu2fdB8hhrClW7t3VDSM6oG7F54=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYRA04Dw64C9164E","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"46K views"}},"simpleText":"46K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"COABEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"COABEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"BF1d6A18AFM","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"COABEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["BF1d6A18AFM"],"params":"CAQ%3D"}},"videoIds":["BF1d6A18AFM"]}},{"clickTrackingParams":"COABEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"COEBELlqIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},"popupType":"TOAST"}}]}},"trackingParams":"COABEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtCRjFkNkExOEFGTQ%3D%3D","commands":[{"clickTrackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CN8BEI5iIhMI9-mLjPv1iwMVUnR6BR3L-BGE","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","hasSeparator":true}}],"trackingParams":"CNwBEKQwGAAiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","accessibility":{"accessibilityData":{"label":"Action menu"}},"targetId":"watch-related-menu-button"}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"42 minutes, 15 seconds"}},"simpleText":"42:15"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CN4BEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"BF1d6A18AFM","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CN4BEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"BF1d6A18AFM"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CN4BEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CN0BEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CN0BEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"BF1d6A18AFM","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CN0BEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["BF1d6A18AFM"],"params":"CAQ%3D"}},"videoIds":["BF1d6A18AFM"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CN0BEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"14.6: Directional Derivatives and the Gradient Vector (1/2) - 42 minutes - Go to channel - Alexandra Niedden - 46K views - 5 years ago - play video"}},"richThumbnail":{"movingThumbnailRenderer":{"movingThumbnailDetails":{"thumbnails":[{"url":"https://i.ytimg.com/an_webp/BF1d6A18AFM/mqdefault_6s.webp?du=3000 sqp=COmcp74G rs=AOn4CLB0pOlaSitPOh_0pJeo-2BOFJEFFg","width":320,"height":180}],"logAsMovingThumbnail":true},"enableHoveredLogging":true,"enableOverlay":true}}}},{"compactVideoRenderer":{"videoId":"a-a7FaA7voQ","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/a-a7FaA7voQ/hqdefault.jpg?sqp=-oaymwE1CKgBEF5IVfKriqkDKAgBFQAAiEIYAXABwAEG8AEB-AG-B4AC0AWKAgwIABABGFYgXyhlMA8= rs=AOn4CLBv3uGj5y8l2VA8hzTZE84lI3iRHg","width":168,"height":94},{"url":"https://i.ytimg.com/vi/a-a7FaA7voQ/hqdefault.jpg?sqp=-oaymwE2CNACELwBSFXyq4qpAygIARUAAIhCGAFwAcABBvABAfgBvgeAAtAFigIMCAAQARhWIF8oZTAP rs=AOn4CLDEJWWnq2XY-uSilkApjpOhhSHcog","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"14.3: Partial Derivatives by Alexandra Niedden 30,170 views 2 years ago 31 minutes"}},"simpleText":"14.3: Partial Derivatives"},"longBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"publishedTimeText":{"simpleText":"2 years ago"},"viewCountText":{"simpleText":"30,170 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"31 minutes, 51 seconds"}},"simpleText":"31:51"},"navigationEndpoint":{"clickTrackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWRIgs-ngY_u3aGTAZoBBQgBEPgd","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=a-a7FaA7voQ","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"a-a7FaA7voQ","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 id=6be6bb15a03bbe84 ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1643750 mt=1741281175 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_n4nO9hINEkA9rVfShhbu2fdB8hhrClW7t3VDSM6oG7F54=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYRAhP3ugdrirvNr","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"30K views"}},"simpleText":"30K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CNoBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CNoBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"a-a7FaA7voQ","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CNoBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["a-a7FaA7voQ"],"params":"CAQ%3D"}},"videoIds":["a-a7FaA7voQ"]}},{"clickTrackingParams":"CNoBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CNsBELlqIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},"popupType":"TOAST"}}]}},"trackingParams":"CNoBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgthLWE3RmFBN3ZvUQ%3D%3D","commands":[{"clickTrackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CNkBEI5iIhMI9-mLjPv1iwMVUnR6BR3L-BGE","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","hasSeparator":true}}],"trackingParams":"CNYBEKQwGAEiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"31 minutes, 51 seconds"}},"simpleText":"31:51"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CNgBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"a-a7FaA7voQ","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CNgBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"a-a7FaA7voQ"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CNgBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CNcBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CNcBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"a-a7FaA7voQ","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CNcBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["a-a7FaA7voQ"],"params":"CAQ%3D"}},"videoIds":["a-a7FaA7voQ"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CNcBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"14.3: Partial Derivatives - 31 minutes - Go to channel - Alexandra Niedden - 30K views - 2 years ago - play video"}},"richThumbnail":{"movingThumbnailRenderer":{"enableHoveredLogging":true,"enableOverlay":true}}}},{"compactVideoRenderer":{"videoId":"HqkDcfMIbC0","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/HqkDcfMIbC0/hqdefault.jpg?sqp=-oaymwE1CKgBEF5IVfKriqkDKAgBFQAAiEIYAXABwAEG8AEB-AG-B4AC0AWKAgwIABABGEAgXihyMA8= rs=AOn4CLDIA_fiJYU62F95-IBB4gd562uOYQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/HqkDcfMIbC0/hqdefault.jpg?sqp=-oaymwE2CNACELwBSFXyq4qpAygIARUAAIhCGAFwAcABBvABAfgBvgeAAtAFigIMCAAQARhAIF4ocjAP rs=AOn4CLD4PlnIYfXbG2NUfDcHYaqK8cEraA","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"Calc 3 14.8 Notes: Lagrange Multipliers by Alexandra Niedden 7,503 views 10 months ago 28 minutes"}},"simpleText":"Calc 3 14.8 Notes: Lagrange Multipliers"},"longBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"publishedTimeText":{"simpleText":"10 months ago"},"viewCountText":{"simpleText":"7,503 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"28 minutes, 8 seconds"}},"simpleText":"28:08"},"navigationEndpoint":{"clickTrackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWRIgs-ngY_u3aGTAZoBBQgBEPgd","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=HqkDcfMIbC0","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"HqkDcfMIbC0","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr1---sn-nvm-cxb6.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 id=1ea90371f3086c2d ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=883750 mt=1741281175 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_n4nO9hINEkA9rVfShhbu2fdB8hhrClW7t3VDSM6oG7F54=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYRArdihmJ_uwNQe","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"7.5K views"}},"simpleText":"7.5K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CNQBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CNQBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"HqkDcfMIbC0","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CNQBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["HqkDcfMIbC0"],"params":"CAQ%3D"}},"videoIds":["HqkDcfMIbC0"]}},{"clickTrackingParams":"CNQBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CNUBELlqIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},"popupType":"TOAST"}}]}},"trackingParams":"CNQBEP6YBBgFIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtIcWtEY2ZNSWJDMA%3D%3D","commands":[{"clickTrackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CNMBEI5iIhMI9-mLjPv1iwMVUnR6BR3L-BGE","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","hasSeparator":true}}],"trackingParams":"CNABEKQwGAIiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"28 minutes, 8 seconds"}},"simpleText":"28:08"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CNIBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"HqkDcfMIbC0","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CNIBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"HqkDcfMIbC0"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CNIBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CNEBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CNEBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"HqkDcfMIbC0","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CNEBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["HqkDcfMIbC0"],"params":"CAQ%3D"}},"videoIds":["HqkDcfMIbC0"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CNEBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"Calc 3 14.8 Notes: Lagrange Multipliers - 28 minutes - Go to channel - Alexandra Niedden - 7.5K views - 10 months ago - play video"}},"richThumbnail":{"movingThumbnailRenderer":{"enableHoveredLogging":true,"enableOverlay":true}}}},{"compactVideoRenderer":{"videoId":"rmf_vnM3ZIw","thumbnail":{"thumbnails":[{"url":"https://i.ytimg.com/vi/rmf_vnM3ZIw/hqdefault.jpg?sqp=-oaymwEbCKgBEF5IVfKriqkDDggBFQAAiEIYAXABwAEG rs=AOn4CLAhUBN-cOurP1ZRb_g1tVQwep32xQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/rmf_vnM3ZIw/hqdefault.jpg?sqp=-oaymwEcCNACELwBSFXyq4qpAw4IARUAAIhCGAFwAcABBg== rs=AOn4CLCMFX44sTakle18iJyRoPal_wEV-g","width":336,"height":188}]},"title":{"accessibility":{"accessibilityData":{"label":"14.4: Tangent Planes and Linear Approximations (1/2) by Alexandra Niedden 62,747 views 5 years ago 39 minutes"}},"simpleText":"14.4: Tangent Planes and Linear Approximations (1/2)"},"longBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"publishedTimeText":{"simpleText":"5 years ago"},"viewCountText":{"simpleText":"62,747 views"},"lengthText":{"accessibility":{"accessibilityData":{"label":"39 minutes, 55 seconds"}},"simpleText":"39:55"},"navigationEndpoint":{"clickTrackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWRIgs-ngY_u3aGTAZoBBQgBEPgd","commandMetadata":{"webCommandMetadata":{"url":"/watch?v=rmf_vnM3ZIw","webPageType":"WEB_PAGE_TYPE_WATCH","rootVe":3832}},"watchEndpoint":{"videoId":"rmf_vnM3ZIw","nofollow":true,"watchEndpointSupportedOnesieConfig":{"html5PlaybackOnesieConfig":{"commonConfig":{"url":"https://rr3---sn-uxaxiv0nxx5q-nv4l.googlevideo.com/initplayback?source=youtube oeis=1 c=WEB oad=3200 ovd=3200 oaad=11000 oavd=11000 ocs=700 oewis=1 oputc=1 ofpcc=1 msp=1 odepv=1 id=ae67ffbe7337648c ip=2a02%3A27aa%3A0%3A0%3A0%3A0%3A0%3Ad49 initcwndbps=1643750 mt=1741281175 oweuc="}}}}},"shortBylineText":{"runs":[{"text":"Alexandra Niedden","navigationEndpoint":{"clickTrackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQyB3JlbGF0ZWQ=","commandMetadata":{"webCommandMetadata":{"url":"/@alexandraniedden5337","webPageType":"WEB_PAGE_TYPE_CHANNEL","rootVe":3611,"apiUrl":"/youtubei/v1/browse"}},"browseEndpoint":{"browseId":"UCmyeeTlzwtoBOCK9RfG13cQ","canonicalBaseUrl":"/@alexandraniedden5337"}}}]},"channelThumbnail":{"thumbnails":[{"url":"https://yt3.ggpht.com/ytc/AIdro_n4nO9hINEkA9rVfShhbu2fdB8hhrClW7t3VDSM6oG7F54=s68-c-k-c0x00ffffff-no-rj","width":68,"height":68}]},"trackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYRAjMndmef3_7OuAQ==","shortViewCountText":{"accessibility":{"accessibilityData":{"label":"62K views"}},"simpleText":"62K views"},"menu":{"menuRenderer":{"items":[{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Add to queue"}]},"icon":{"iconType":"ADD_TO_QUEUE_TAIL"},"serviceEndpoint":{"clickTrackingParams":"CM4BEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CM4BEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"rmf_vnM3ZIw","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CM4BEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["rmf_vnM3ZIw"],"params":"CAQ%3D"}},"videoIds":["rmf_vnM3ZIw"]}},{"clickTrackingParams":"CM4BEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE","openPopupAction":{"popup":{"notificationActionRenderer":{"responseText":{"simpleText":"Added to queue"},"trackingParams":"CM8BELlqIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},"popupType":"TOAST"}}]}},"trackingParams":"CM4BEP6YBBgGIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"menuServiceItemRenderer":{"text":{"runs":[{"text":"Share"}]},"icon":{"iconType":"SHARE"},"serviceEndpoint":{"clickTrackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/share/get_share_panel"}},"shareEntityServiceEndpoint":{"serializedShareEntity":"CgtybWZfdm5NM1pJdw%3D%3D","commands":[{"clickTrackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","openPopupAction":{"popup":{"unifiedSharePanelRenderer":{"trackingParams":"CM0BEI5iIhMI9-mLjPv1iwMVUnR6BR3L-BGE","showLoadingSpinner":true}},"popupType":"DIALOG","beReused":true}}]}},"trackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","hasSeparator":true}}],"trackingParams":"CMoBEKQwGAMiEwj36YuM-_WLAxVSdHoFHcv4EYQ=","accessibility":{"accessibilityData":{"label":"Action menu"}}}},"thumbnailOverlays":[{"thumbnailOverlayTimeStatusRenderer":{"text":{"accessibility":{"accessibilityData":{"label":"39 minutes, 55 seconds"}},"simpleText":"39:55"},"style":"DEFAULT"}},{"thumbnailOverlayToggleButtonRenderer":{"isToggled":false,"untoggledIcon":{"iconType":"WATCH_LATER"},"toggledIcon":{"iconType":"CHECK"},"untoggledTooltip":"Watch later","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMwBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"addedVideoId":"rmf_vnM3ZIw","action":"ACTION_ADD_VIDEO"}]}},"toggledServiceEndpoint":{"clickTrackingParams":"CMwBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/browse/edit_playlist"}},"playlistEditEndpoint":{"playlistId":"WL","actions":[{"action":"ACTION_REMOVE_VIDEO_BY_VIDEO_ID","removedVideoId":"rmf_vnM3ZIw"}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Watch later"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMwBEPnnAxgBIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayToggleButtonRenderer":{"untoggledIcon":{"iconType":"ADD_TO_QUEUE_TAIL"},"toggledIcon":{"iconType":"PLAYLIST_ADD_CHECK"},"untoggledTooltip":"Add to queue","toggledTooltip":"Added","untoggledServiceEndpoint":{"clickTrackingParams":"CMsBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true}},"signalServiceEndpoint":{"signal":"CLIENT_SIGNAL","actions":[{"clickTrackingParams":"CMsBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","addToPlaylistCommand":{"openMiniplayer":false,"openListPanel":true,"videoId":"rmf_vnM3ZIw","listType":"PLAYLIST_EDIT_LIST_TYPE_QUEUE","onCreateListCommand":{"clickTrackingParams":"CMsBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE","commandMetadata":{"webCommandMetadata":{"sendPost":true,"apiUrl":"/youtubei/v1/playlist/create"}},"createPlaylistServiceEndpoint":{"videoIds":["rmf_vnM3ZIw"],"params":"CAQ%3D"}},"videoIds":["rmf_vnM3ZIw"]}}]}},"untoggledAccessibility":{"accessibilityData":{"label":"Add to queue"}},"toggledAccessibility":{"accessibilityData":{"label":"Added"}},"trackingParams":"CMsBEMfsBBgCIhMI9-mLjPv1iwMVUnR6BR3L-BGE"}},{"thumbnailOverlayNowPlayingRenderer":{"text":{"runs":[{"text":"Now playing"}]}}}],"accessibility":{"accessibilityData":{"label":"14.4: Tangent Planes and Linear Approximations (1/2) - 39 minutes - Go to channel - Alexandra Niedden - 62K views - 5 years ago - play video"}},"richThumbnail":{"movingThumbnailRenderer":{"movingThumbnailDetails":{"thumbnails":[{"url":"https://i.ytimg.com/an_webp/rmf_vnM3ZIw/mqdefault_6s.webp?du=3000 sqp=COT5pr4G rs=AOn4CLBKq-QkN5RtXzJpGeVFk4nwK-PY-w","width":320,"height":180}],"logAsMovingThumbnail":true},"enableHoveredLogging":true,"enableOverlay":true}}}},{"lockupViewModel":{"contentImage":{"collectionThumbnailViewModel":{"primaryThumbnail":{"thumbnailViewModel":{"image":{"sources":[{"url":"https://i.ytimg.com/vi/4SbxluFD47U/hqdefault.jpg?sqp=-oaymwEWCKgBEF5IWvKriqkDCQgBFQAAiEIYAQ== rs=AOn4CLClUkj0RYjzWXDtSoLhebL2XkmpcQ","width":168,"height":94},{"url":"https://i.ytimg.com/vi/4SbxluFD47U/hqdefault.jpg?sqp=-oaymwEXCNACELwBSFryq4qpAwkIARUAAIhCGAE= rs=AOn4CLDaUImm7NdzI3WMgK82axFbzd79wA","width":336,"height":188}]},"overlays":[{"thumbnailOverlayBadgeViewModel":{"thumbnailBadges":[{"thumbnailBadgeViewModel":{"icon":{"sources":[{"clientResource":{"imageName":"PLAYLISTS"}}]},"text":"10 videos","badgeStyle":"THUMBNAIL_OVERLAY_BADGE_STYLE_DEFAULT","backgroundColor":{"lightTheme":3351062,"darkTheme":3351062}}}],"position":"THUMBNAIL_OVERLAY_BADGE_POSITION_BOTTOM_END"}},{"thumbnailHoverOverlayViewModel":{"icon":{"sources":[{"clientResource":{"imageName":"PLAY_ALL"}}]},"text":{"content":"Play all","styleRuns":[{"startIndex":0,"length":8}]},"style":"THUMBNAIL_HOVER_OVERLAY_STYLE_COVER"}}],"backgroundColor":{"lightTheme":4993826,"darkTheme":4993826}}},"stackColor":{"lightTheme":10059371,"darkTheme":9667190}}},"metadata":{"lockupMetadataViewModel":{"title":{"content":"Ch 14: Partial Derivatives"},"metadata":{"contentMetadataViewModel":{"metadataRows":[{"metadataParts":[{"text":{"content":"Alexandra Niedden

#############################

New on site