{"id":235,"date":"2023-09-20T22:48:28","date_gmt":"2023-09-20T22:48:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/graphing-a-derivative\/"},"modified":"2024-08-05T12:36:03","modified_gmt":"2024-08-05T12:36:03","slug":"the-derivative-as-a-function-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-derivative-as-a-function-learn-it-2\/","title":{"raw":"The Derivative as a Function: Learn It 2","rendered":"The Derivative as a Function: Learn It 2"},"content":{"raw":"<h2>Graphing a Derivative<\/h2>\r\n<p id=\"fs-id1169737952664\">Given the equation of a function or its derivative, we can graph it to understand the relationship between the two. The derivative [latex]f^{\\prime}(x)[\/latex] gives the rate of change of a function [latex]f(x)[\/latex] or the slope of the tangent line to [latex]f(x)[\/latex]. To understand this relationship better, it is helpful to recall the characteristics of lines with certain slopes.<\/p>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li><strong>Positive Slope<\/strong>: A line has a positive slope if it is increasing from left to right.<\/li>\r\n\t<li><strong>Negative Slope<\/strong>: A line has a negative slope if it is decreasing from left to right.<\/li>\r\n\t<li><strong>Zero Slope<\/strong>: A horizontal line has a slope of [latex]0[\/latex].<\/li>\r\n\t<li><strong>Undefined Slope<\/strong>: A vertical line has an undefined slope.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p id=\"fs-id1169738190458\">In the first example on the previous page, we found that for [latex]f(x)=\\sqrt{x}, \\, f^{\\prime}(x)=\\frac{1}{2}\\sqrt{x}[\/latex]. If we graph these functions on the same axes, as in Figure 2, we can use the graphs to understand the relationship between these two functions.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205223\/CNX_Calc_Figure_03_02_002.jpg\" alt=\"The function f(x) = the square root of x is graphed as is its derivative f\u2019(x) = 1\/(2 times the square root of x).\" width=\"487\" height=\"322\" \/> Figure 2. The derivative [latex]f^{\\prime}(x)[\/latex] is positive everywhere because the function [latex]f(x)[\/latex] is increasing.[\/caption]\r\n\r\n<p>Looking at the graphs, notice that [latex]f(x)[\/latex] is increasing over its entire domain, meaning the slopes of its tangent lines at all points are positive. Consequently, [latex]f^{\\prime}(x)&gt;0[\/latex] for all values of [latex]x[\/latex] in its domain. As [latex]x[\/latex] increases, the slopes of the tangent lines to [latex]f(x)[\/latex] decrease, leading to a corresponding decrease in [latex]f^{\\prime}(x)[\/latex]. Additionally, [latex]f(0)[\/latex] is undefined and that [latex]\\underset{x\\to 0^+}{\\lim}f^{\\prime}(x)=+\\infty[\/latex], corresponding to a vertical tangent to [latex]f(x)[\/latex] at [latex]0[\/latex].<\/p>\r\n<p id=\"fs-id1169738183946\">In the second example, we found that for [latex]f(x)=x^2-2x, \\, f^{\\prime}(x)=2x-2[\/latex]. The graphs of these functions are shown in Figure 3.\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205225\/CNX_Calc_Figure_03_02_003.jpg\" alt=\"The function f(x) = x squared \u2013 2x is graphed as is its derivative f\u2019(x) = 2x \u2212 2.\" width=\"487\" height=\"358\" \/> Figure 3. The derivative [latex]f^{\\prime}(x)&lt;0[\/latex] where the function [latex]f(x)[\/latex] is decreasing and [latex]f^{\\prime}(x)&gt;0[\/latex] where [latex]f(x)[\/latex] is increasing. The derivative is zero where the function has a horizontal tangent.[\/caption]\r\n\r\n<p>Observe that [latex]f(x)[\/latex] is decreasing for [latex]x&lt;1[\/latex]. For these values of [latex]x, \\, f^{\\prime}(x)&lt;0[\/latex]. For [latex]x&gt;1, \\, f(x)[\/latex] is increasing and [latex]f^{\\prime}(x)&gt;0[\/latex]. Also, [latex]f(x)[\/latex] has a horizontal tangent at [latex]x=1[\/latex] and [latex]f^{\\prime}(1)=0[\/latex].<\/p>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738187865\">Use the following graph of [latex]f(x)[\/latex] to sketch a graph of [latex]f^{\\prime}(x)[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205228\/CNX_Calc_Figure_03_02_004.jpg\" alt=\"The function f(x) is roughly sinusoidal, starting at (\u22124, 3), decreasing to a local minimum at (\u22122, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2).\" width=\"487\" height=\"471\" \/> Figure 4. Graph of [latex]f(x)[\/latex].[\/caption]\r\n\r\n<p>[reveal-answer q=\"fs-id1169738226318\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169738226318\"]<\/p>\r\n<p id=\"fs-id1169738226318\">The solution is shown in the following graph. Observe that [latex]f(x)[\/latex] is increasing and [latex]f^{\\prime}(x)&gt;0[\/latex] on [latex](\u20132,3)[\/latex]. Also, [latex]f(x)[\/latex] is decreasing and [latex]f^{\\prime}(x)&lt;0[\/latex] on [latex](\u2212\\infty ,-2)[\/latex] and on [latex](3,+\\infty)[\/latex]. Also note that [latex]f(x)[\/latex] has horizontal tangents at [latex]-2[\/latex] and [latex]3[\/latex], and [latex]f^{\\prime}(-2)=0[\/latex] and [latex]f^{\\prime}(3)=0[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205232\/CNX_Calc_Figure_03_02_005.jpg\" alt=\"Two functions are graphed here: f(x) and f\u2019(x). The function f(x) is the same as the above graph, that is, roughly sinusoidal, starting at (\u22124, 3), decreasing to a local minimum at (\u22122, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2). The function f\u2019(x) is an downward-facing parabola with vertex near (0.5, 1.75), y-intercept (0, 1.5), and x-intercepts (\u22121.9, 0) and (3, 0).\" width=\"487\" height=\"471\" \/> Figure 5. Graph of\u00a0[latex]f^{\\prime}(x)[\/latex].[\/caption]\r\n\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/w9m4h4i47-M?controls=0&amp;start=415&amp;end=535&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction415to535_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.2 The Derivative as a Function\" here (opens in new window)<\/a>.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288380[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<h2>Graphing a Derivative<\/h2>\n<p id=\"fs-id1169737952664\">Given the equation of a function or its derivative, we can graph it to understand the relationship between the two. The derivative [latex]f^{\\prime}(x)[\/latex] gives the rate of change of a function [latex]f(x)[\/latex] or the slope of the tangent line to [latex]f(x)[\/latex]. To understand this relationship better, it is helpful to recall the characteristics of lines with certain slopes.<\/p>\n<section class=\"textbox recall\">\n<ul>\n<li><strong>Positive Slope<\/strong>: A line has a positive slope if it is increasing from left to right.<\/li>\n<li><strong>Negative Slope<\/strong>: A line has a negative slope if it is decreasing from left to right.<\/li>\n<li><strong>Zero Slope<\/strong>: A horizontal line has a slope of [latex]0[\/latex].<\/li>\n<li><strong>Undefined Slope<\/strong>: A vertical line has an undefined slope.<\/li>\n<\/ul>\n<\/section>\n<p id=\"fs-id1169738190458\">In the first example on the previous page, we found that for [latex]f(x)=\\sqrt{x}, \\, f^{\\prime}(x)=\\frac{1}{2}\\sqrt{x}[\/latex]. If we graph these functions on the same axes, as in Figure 2, we can use the graphs to understand the relationship between these two functions.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205223\/CNX_Calc_Figure_03_02_002.jpg\" alt=\"The function f(x) = the square root of x is graphed as is its derivative f\u2019(x) = 1\/(2 times the square root of x).\" width=\"487\" height=\"322\" \/><figcaption class=\"wp-caption-text\">Figure 2. The derivative [latex]f^{\\prime}(x)[\/latex] is positive everywhere because the function [latex]f(x)[\/latex] is increasing.<\/figcaption><\/figure>\n<p>Looking at the graphs, notice that [latex]f(x)[\/latex] is increasing over its entire domain, meaning the slopes of its tangent lines at all points are positive. Consequently, [latex]f^{\\prime}(x)>0[\/latex] for all values of [latex]x[\/latex] in its domain. As [latex]x[\/latex] increases, the slopes of the tangent lines to [latex]f(x)[\/latex] decrease, leading to a corresponding decrease in [latex]f^{\\prime}(x)[\/latex]. Additionally, [latex]f(0)[\/latex] is undefined and that [latex]\\underset{x\\to 0^+}{\\lim}f^{\\prime}(x)=+\\infty[\/latex], corresponding to a vertical tangent to [latex]f(x)[\/latex] at [latex]0[\/latex].<\/p>\n<p id=\"fs-id1169738183946\">In the second example, we found that for [latex]f(x)=x^2-2x, \\, f^{\\prime}(x)=2x-2[\/latex]. The graphs of these functions are shown in Figure 3.\u00a0<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205225\/CNX_Calc_Figure_03_02_003.jpg\" alt=\"The function f(x) = x squared \u2013 2x is graphed as is its derivative f\u2019(x) = 2x \u2212 2.\" width=\"487\" height=\"358\" \/><figcaption class=\"wp-caption-text\">Figure 3. The derivative [latex]f^{\\prime}(x)&lt;0[\/latex] where the function [latex]f(x)[\/latex] is decreasing and [latex]f^{\\prime}(x)&gt;0[\/latex] where [latex]f(x)[\/latex] is increasing. The derivative is zero where the function has a horizontal tangent.<\/figcaption><\/figure>\n<p>Observe that [latex]f(x)[\/latex] is decreasing for [latex]x<1[\/latex]. For these values of [latex]x, \\, f^{\\prime}(x)<0[\/latex]. For [latex]x>1, \\, f(x)[\/latex] is increasing and [latex]f^{\\prime}(x)>0[\/latex]. Also, [latex]f(x)[\/latex] has a horizontal tangent at [latex]x=1[\/latex] and [latex]f^{\\prime}(1)=0[\/latex].<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738187865\">Use the following graph of [latex]f(x)[\/latex] to sketch a graph of [latex]f^{\\prime}(x)[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205228\/CNX_Calc_Figure_03_02_004.jpg\" alt=\"The function f(x) is roughly sinusoidal, starting at (\u22124, 3), decreasing to a local minimum at (\u22122, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2).\" width=\"487\" height=\"471\" \/><figcaption class=\"wp-caption-text\">Figure 4. Graph of [latex]f(x)[\/latex].<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169738226318\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169738226318\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738226318\">The solution is shown in the following graph. Observe that [latex]f(x)[\/latex] is increasing and [latex]f^{\\prime}(x)>0[\/latex] on [latex](\u20132,3)[\/latex]. Also, [latex]f(x)[\/latex] is decreasing and [latex]f^{\\prime}(x)<0[\/latex] on [latex](\u2212\\infty ,-2)[\/latex] and on [latex](3,+\\infty)[\/latex]. Also note that [latex]f(x)[\/latex] has horizontal tangents at [latex]-2[\/latex] and [latex]3[\/latex], and [latex]f^{\\prime}(-2)=0[\/latex] and [latex]f^{\\prime}(3)=0[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205232\/CNX_Calc_Figure_03_02_005.jpg\" alt=\"Two functions are graphed here: f(x) and f\u2019(x). The function f(x) is the same as the above graph, that is, roughly sinusoidal, starting at (\u22124, 3), decreasing to a local minimum at (\u22122, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2). The function f\u2019(x) is an downward-facing parabola with vertex near (0.5, 1.75), y-intercept (0, 1.5), and x-intercepts (\u22121.9, 0) and (3, 0).\" width=\"487\" height=\"471\" \/><figcaption class=\"wp-caption-text\">Figure 5. Graph of\u00a0[latex]f^{\\prime}(x)[\/latex].<\/figcaption><\/figure>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/w9m4h4i47-M?controls=0&amp;start=415&amp;end=535&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction415to535_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.2 The Derivative as a Function&#8221; here (opens in new window)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288380\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288380&theme=lumen&iframe_resize_id=ohm288380&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.2 The Derivative as a Function\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"3.2 The Derivative as a Function","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/235"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/235\/revisions"}],"predecessor-version":[{"id":4494,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/235\/revisions\/4494"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/235\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=235"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=235"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=235"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}