{"id":341,"date":"2021-02-04T01:15:19","date_gmt":"2021-02-04T01:15:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=341"},"modified":"2022-03-16T05:31:29","modified_gmt":"2022-03-16T05:31:29","slug":"combining-differentiation-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/combining-differentiation-rules\/","title":{"raw":"Combining Differentiation Rules","rendered":"Combining Differentiation Rules"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Combine the differentiation rules to find the derivative of a polynomial or rational function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1169739252090\">As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.<\/p>\r\n\r\n<div id=\"fs-id1169739347062\" class=\"textbook exercises\">\r\n<h3>Example: Combining Differentiation Rules<\/h3>\r\n<p id=\"fs-id1169739347072\">For [latex]k(x)=3h(x)+x^2g(x)[\/latex], find [latex]k^{\\prime}(x)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739347144\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739347144\"]\r\n<p id=\"fs-id1169739347144\">Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.<\/p>\r\n\r\n<div id=\"fs-id1169739347147\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) &amp; =\\frac{d}{dx}(3h(x)+x^2g(x))=\\frac{d}{dx}(3h(x))+\\frac{d}{dx}(x^2g(x)) &amp; &amp; &amp; \\text{Apply the sum rule.} \\\\ &amp; =3\\frac{d}{dx}(h(x))+(\\frac{d}{dx}(x^2)g(x)+\\frac{d}{dx}(g(x))x^2) &amp; &amp; &amp; \\begin{array}{l}\\text{Apply the constant multiple rule to} \\\\ \\text{differentiate} \\, 3h(x) \\, \\text{and the product} \\\\ \\text{rule to differentiate} \\, x^2g(x). \\end{array} \\\\ &amp; =3h^{\\prime}(x)+2xg(x)+g^{\\prime}(x)x^2 &amp; &amp; &amp; \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739325719\" class=\"textbook exercises\">\r\n<h3>Example: Extending the Product Rule<\/h3>\r\n<p id=\"fs-id1169739325728\">For [latex]k(x)=f(x)g(x)h(x)[\/latex], express [latex]k^{\\prime}(x)[\/latex] in terms of [latex]f(x), \\, g(x), \\, h(x)[\/latex], and their derivatives.<\/p>\r\n[reveal-answer q=\"fs-id1169739270350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739270350\"]\r\n<p id=\"fs-id1169739270350\">We can think of the function [latex]k(x)[\/latex] as the product of the function [latex]f(x)g(x)[\/latex] and the function [latex]h(x)[\/latex]. That is, [latex]k(x)=(f(x)g(x))\\cdot h(x)[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1169739333852\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) &amp; =\\frac{d}{dx}(f(x)g(x))\\cdot h(x)+\\frac{d}{dx}(h(x))\\cdot (f(x)g(x)) &amp; &amp; &amp; \\begin{array}{l}\\text{Apply the product rule to the product} \\\\ \\text{of} \\, f(x)g(x) \\, \\text{and} \\, h(x). \\end{array} \\\\ &amp; =(f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x))h(x)+h^{\\prime}(x)f(x)g(x) &amp; &amp; &amp; \\text{Apply the product rule to} \\, f(x)g(x). \\\\ &amp; =f^{\\prime}(x)g(x)h(x)+f(x)g^{\\prime}(x)h(x)+f(x)g(x)h^{\\prime}(x). &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Extending the Product Rule.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1225&amp;end=1359&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]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.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1225to1359_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169736658392\" class=\"textbook exercises\">\r\n<h3>Example: Combining the Quotient Rule and the Product Rule<\/h3>\r\n<p id=\"fs-id1169736658401\">For [latex]h(x)=\\large \\frac{2x^3k(x)}{3x+2}[\/latex], find [latex]h^{\\prime}(x)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736658474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736658474\"]\r\n<p id=\"fs-id1169736658474\">This procedure is typical for finding the derivative of a rational function.<\/p>\r\n\r\n<div id=\"fs-id1169736658477\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}h^{\\prime}(x) &amp; =\\large \\frac{\\frac{d}{dx}(2x^3k(x))\\cdot (3x+2)-\\frac{d}{dx}(3x+2)\\cdot (2x^3k(x))}{(3x+2)^2} &amp; &amp; &amp; \\text{Apply the quotient rule.} \\\\ &amp; =\\large \\frac{(6x^2k(x)+k^{\\prime}(x)\\cdot 2x^3)(3x+2)-3(2x^3k(x))}{(3x+2)^2} &amp; &amp; &amp; \\begin{array}{l}\\text{Apply the product rule to find} \\\\ \\frac{d}{dx}(2x^3k(x)). \\, \\text{Use} \\, \\frac{d}{dx}(3x+2)=3. \\end{array} \\\\ &amp; =\\large \\frac{-6x^3k(x)+18x^3k(x)+12x^2k(x)+6x^4k^{\\prime}(x)+4x^3k^{\\prime}(x)}{(3x+2)^2} &amp; &amp; &amp; \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736607611\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169736607620\">Find [latex]\\frac{d}{dx}(3f(x)-2g(x))[\/latex].<\/p>\r\n[reveal-answer q=\"288744\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"288744\"]\r\n<p id=\"fs-id1169736589229\">Apply the difference rule and the constant multiple rule.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736607671\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736607671\"]\r\n<p id=\"fs-id1169736607671\">[latex]3f^{\\prime}(x)-2g^{\\prime}(x)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169736589236\" class=\"textbook exercises\">\r\n<h3>Example: Determining Where a Function Has a Horizontal Tangent<\/h3>\r\n<p id=\"fs-id1169736589245\">Determine the values of [latex]x[\/latex] for which [latex]f(x)=x^3-7x^2+8x+1[\/latex] has a horizontal tangent line.<\/p>\r\n[reveal-answer q=\"fs-id1169736589298\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736589298\"]\r\n<p id=\"fs-id1169736589298\">To find the values of [latex]x[\/latex] for which [latex]f(x)[\/latex] has a horizontal tangent line, we must solve [latex]f^{\\prime}(x)=0[\/latex]. Since<\/p>\r\n\r\n<div id=\"fs-id1169736589343\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3x^2-14x+8=(3x-2)(x-4)[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739111144\">we must solve [latex](3x-2)(x-4)=0[\/latex]. Thus we see that the function has horizontal tangent lines at [latex]x=\\frac{2}{3}[\/latex] and [latex]x=4[\/latex] as shown in the following graph.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"379\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205334\/CNX_Calc_Figure_03_03_002.jpg\" alt=\"The graph shows f(x) = x3 \u2013 7x2 + 8x + 1, and the tangent lines are shown as x = 2\/3 and x = 4.\" width=\"379\" height=\"310\" \/> Figure 2. This function has horizontal tangent lines at [latex]x = 2\/3[\/latex] and [latex]x = 4[\/latex].[\/caption]\r\n<div class=\"wp-caption-text\"><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Determining Where a Function Has a Horizontal Tangent.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1476&amp;end=1576&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]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.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.3DifferentiationRules1476to1576_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Differentiation Rules\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739281977\" class=\"textbook exercises\">\r\n<h3>Example: Finding a Velocity<\/h3>\r\n<p id=\"fs-id1169739281986\">The position of an object on a coordinate axis at time [latex]t[\/latex] is given by [latex]s(t)=\\dfrac{t}{t^2+1}[\/latex]. What is the initial velocity of the object?<\/p>\r\n[reveal-answer q=\"fs-id1169739282028\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739282028\"]\r\n<p id=\"fs-id1169739282028\">Since the initial velocity is [latex]v(0)=s^{\\prime}(0)[\/latex], begin by finding [latex]s^{\\prime}(t)[\/latex] by applying the quotient rule:<\/p>\r\n\r\n<div id=\"fs-id1169739282080\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]s^{\\prime}(t)=\\dfrac{1(t^2+1)-2t(t)}{(t^2+1)^2}=\\dfrac{1-t^2}{(t^2+1)^2}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739301434\">After evaluating, we see that [latex]v(0)=1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739301458\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739301467\">Find the value(s) of [latex]x[\/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[\/latex] is parallel to the line [latex]y=2x+3[\/latex].<\/p>\r\n[reveal-answer q=\"825443\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"825443\"]\r\n<p id=\"fs-id1169739298001\">Solve the equation [latex]f^{\\prime}(x)=2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739297983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739297983\"]\r\n<p id=\"fs-id1169739297983\">[latex]\\frac{5}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]33700[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739298028\" class=\"textbox tryit\">\r\n<h3>Activity: Racetrack Safety at the Formula One Grandstand<\/h3>\r\n<p id=\"fs-id1169739298036\">Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (Figure 3).<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"900\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205340\/CNX_Calc_Figure_03_03_003.jpg\" alt=\"A photo of a grandstand next to a straightaway of a race track.\" width=\"900\" height=\"415\" \/> Figure 3. The grandstand next to a straightaway of the Circuit de Barcelona-Catalunya race track, located where the spectators are not in danger.[\/caption]\r\n\r\n&nbsp;\r\n<p id=\"fs-id1169739298067\">Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack.<\/p>\r\n<p id=\"fs-id1169739298074\">Suppose you are designing a new Formula One track. One section of the track can be modeled by the function [latex]f(x)=x^3+3x^2+x[\/latex] (Figure 4). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. The plans call for the front corner of the grandstand to be located at the point [latex](-1.9,2.8)[\/latex]. We want to determine whether this location puts the spectators in danger if a driver loses control of the car.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"860\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205343\/CNX_Calc_Figure_03_03_004.jpg\" alt=\"This figure has two parts labeled a and b. Figure a shows the graph of f(x) = x3 + 3x2 + x. Figure b shows the same graph but this time with two boxes on it. The first box appears along the left-hand side of the graph straddling the x-axis roughly parallel to f(x). The second box appears a little higher, also roughly parallel to f(x), with its front corner located at (\u22121.9, 2.8). Note that this corner is roughly in line with the direct path of the track before it started to turn.\" width=\"860\" height=\"462\" \/> Figure 4. (a) One section of the racetrack can be modeled by the function [latex]f(x)=x^3+3x^2+x[\/latex]. (b) The front corner of the grandstand is located at [latex](-1.9,2.8)[\/latex].[\/caption]\r\n<div class=\"wp-caption-text\"><\/div>\r\n<ol id=\"fs-id1169736655867\">\r\n \t<li>Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the [latex](x,y)[\/latex] coordinates of this point near the turn.<\/li>\r\n \t<li>Find the equation of the tangent line to the curve at this point.<\/li>\r\n \t<li>To determine whether the spectators are in danger in this scenario, find the [latex]x[\/latex]-coordinate of the point where the tangent line crosses the line [latex]y=2.8[\/latex]. Is this point safely to the right of the grandstand? Or are the spectators in danger?<\/li>\r\n \t<li>What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point [latex](-2.5,0.625)[\/latex]. What is the slope of the tangent line at this point?<\/li>\r\n \t<li>If a driver loses control as described in part 4, are the spectators safe?<\/li>\r\n \t<li>Should you proceed with the current design for the grandstand, or should the grandstands be moved?<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Combine the differentiation rules to find the derivative of a polynomial or rational function<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1169739252090\">As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.<\/p>\n<div id=\"fs-id1169739347062\" class=\"textbook exercises\">\n<h3>Example: Combining Differentiation Rules<\/h3>\n<p id=\"fs-id1169739347072\">For [latex]k(x)=3h(x)+x^2g(x)[\/latex], find [latex]k^{\\prime}(x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739347144\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739347144\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739347144\">Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.<\/p>\n<div id=\"fs-id1169739347147\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) & =\\frac{d}{dx}(3h(x)+x^2g(x))=\\frac{d}{dx}(3h(x))+\\frac{d}{dx}(x^2g(x)) & & & \\text{Apply the sum rule.} \\\\ & =3\\frac{d}{dx}(h(x))+(\\frac{d}{dx}(x^2)g(x)+\\frac{d}{dx}(g(x))x^2) & & & \\begin{array}{l}\\text{Apply the constant multiple rule to} \\\\ \\text{differentiate} \\, 3h(x) \\, \\text{and the product} \\\\ \\text{rule to differentiate} \\, x^2g(x). \\end{array} \\\\ & =3h^{\\prime}(x)+2xg(x)+g^{\\prime}(x)x^2 & & & \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739325719\" class=\"textbook exercises\">\n<h3>Example: Extending the Product Rule<\/h3>\n<p id=\"fs-id1169739325728\">For [latex]k(x)=f(x)g(x)h(x)[\/latex], express [latex]k^{\\prime}(x)[\/latex] in terms of [latex]f(x), \\, g(x), \\, h(x)[\/latex], and their derivatives.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739270350\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739270350\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739270350\">We can think of the function [latex]k(x)[\/latex] as the product of the function [latex]f(x)g(x)[\/latex] and the function [latex]h(x)[\/latex]. That is, [latex]k(x)=(f(x)g(x))\\cdot h(x)[\/latex]. Thus,<\/p>\n<div id=\"fs-id1169739333852\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}k^{\\prime}(x) & =\\frac{d}{dx}(f(x)g(x))\\cdot h(x)+\\frac{d}{dx}(h(x))\\cdot (f(x)g(x)) & & & \\begin{array}{l}\\text{Apply the product rule to the product} \\\\ \\text{of} \\, f(x)g(x) \\, \\text{and} \\, h(x). \\end{array} \\\\ & =(f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x))h(x)+h^{\\prime}(x)f(x)g(x) & & & \\text{Apply the product rule to} \\, f(x)g(x). \\\\ & =f^{\\prime}(x)g(x)h(x)+f(x)g^{\\prime}(x)h(x)+f(x)g(x)h^{\\prime}(x). & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Extending the Product Rule.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1225&amp;end=1359&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">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\/3.3DifferentiationRules1225to1359_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169736658392\" class=\"textbook exercises\">\n<h3>Example: Combining the Quotient Rule and the Product Rule<\/h3>\n<p id=\"fs-id1169736658401\">For [latex]h(x)=\\large \\frac{2x^3k(x)}{3x+2}[\/latex], find [latex]h^{\\prime}(x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736658474\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736658474\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736658474\">This procedure is typical for finding the derivative of a rational function.<\/p>\n<div id=\"fs-id1169736658477\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lllll}h^{\\prime}(x) & =\\large \\frac{\\frac{d}{dx}(2x^3k(x))\\cdot (3x+2)-\\frac{d}{dx}(3x+2)\\cdot (2x^3k(x))}{(3x+2)^2} & & & \\text{Apply the quotient rule.} \\\\ & =\\large \\frac{(6x^2k(x)+k^{\\prime}(x)\\cdot 2x^3)(3x+2)-3(2x^3k(x))}{(3x+2)^2} & & & \\begin{array}{l}\\text{Apply the product rule to find} \\\\ \\frac{d}{dx}(2x^3k(x)). \\, \\text{Use} \\, \\frac{d}{dx}(3x+2)=3. \\end{array} \\\\ & =\\large \\frac{-6x^3k(x)+18x^3k(x)+12x^2k(x)+6x^4k^{\\prime}(x)+4x^3k^{\\prime}(x)}{(3x+2)^2} & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736607611\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169736607620\">Find [latex]\\frac{d}{dx}(3f(x)-2g(x))[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q288744\">Hint<\/span><\/p>\n<div id=\"q288744\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736589229\">Apply the difference rule and the constant multiple rule.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736607671\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736607671\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736607671\">[latex]3f^{\\prime}(x)-2g^{\\prime}(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169736589236\" class=\"textbook exercises\">\n<h3>Example: Determining Where a Function Has a Horizontal Tangent<\/h3>\n<p id=\"fs-id1169736589245\">Determine the values of [latex]x[\/latex] for which [latex]f(x)=x^3-7x^2+8x+1[\/latex] has a horizontal tangent line.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169736589298\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169736589298\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736589298\">To find the values of [latex]x[\/latex] for which [latex]f(x)[\/latex] has a horizontal tangent line, we must solve [latex]f^{\\prime}(x)=0[\/latex]. Since<\/p>\n<div id=\"fs-id1169736589343\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3x^2-14x+8=(3x-2)(x-4)[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739111144\">we must solve [latex](3x-2)(x-4)=0[\/latex]. Thus we see that the function has horizontal tangent lines at [latex]x=\\frac{2}{3}[\/latex] and [latex]x=4[\/latex] as shown in the following graph.<\/p>\n<div style=\"width: 389px\" 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\/11205334\/CNX_Calc_Figure_03_03_002.jpg\" alt=\"The graph shows f(x) = x3 \u2013 7x2 + 8x + 1, and the tangent lines are shown as x = 2\/3 and x = 4.\" width=\"379\" height=\"310\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. This function has horizontal tangent lines at [latex]x = 2\/3[\/latex] and [latex]x = 4[\/latex].<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Determining Where a Function Has a Horizontal Tangent.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ruACLHzWT3g?controls=0&amp;start=1476&amp;end=1576&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">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\/3.3DifferentiationRules1476to1576_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Differentiation Rules&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739281977\" class=\"textbook exercises\">\n<h3>Example: Finding a Velocity<\/h3>\n<p id=\"fs-id1169739281986\">The position of an object on a coordinate axis at time [latex]t[\/latex] is given by [latex]s(t)=\\dfrac{t}{t^2+1}[\/latex]. What is the initial velocity of the object?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739282028\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739282028\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739282028\">Since the initial velocity is [latex]v(0)=s^{\\prime}(0)[\/latex], begin by finding [latex]s^{\\prime}(t)[\/latex] by applying the quotient rule:<\/p>\n<div id=\"fs-id1169739282080\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]s^{\\prime}(t)=\\dfrac{1(t^2+1)-2t(t)}{(t^2+1)^2}=\\dfrac{1-t^2}{(t^2+1)^2}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739301434\">After evaluating, we see that [latex]v(0)=1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739301458\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739301467\">Find the value(s) of [latex]x[\/latex] for which the line tangent to the graph of [latex]f(x)=4x^2-3x+2[\/latex] is parallel to the line [latex]y=2x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q825443\">Hint<\/span><\/p>\n<div id=\"q825443\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739298001\">Solve the equation [latex]f^{\\prime}(x)=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739297983\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739297983\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739297983\">[latex]\\frac{5}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm33700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=33700&theme=oea&iframe_resize_id=ohm33700&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1169739298028\" class=\"textbox tryit\">\n<h3>Activity: Racetrack Safety at the Formula One Grandstand<\/h3>\n<p id=\"fs-id1169739298036\">Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (Figure 3).<\/p>\n<div style=\"width: 910px\" 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\/11205340\/CNX_Calc_Figure_03_03_003.jpg\" alt=\"A photo of a grandstand next to a straightaway of a race track.\" width=\"900\" height=\"415\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. The grandstand next to a straightaway of the Circuit de Barcelona-Catalunya race track, located where the spectators are not in danger.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739298067\">Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack.<\/p>\n<p id=\"fs-id1169739298074\">Suppose you are designing a new Formula One track. One section of the track can be modeled by the function [latex]f(x)=x^3+3x^2+x[\/latex] (Figure 4). The current plan calls for grandstands to be built along the first straightaway and around a portion of the first curve. The plans call for the front corner of the grandstand to be located at the point [latex](-1.9,2.8)[\/latex]. We want to determine whether this location puts the spectators in danger if a driver loses control of the car.<\/p>\n<div style=\"width: 870px\" 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\/11205343\/CNX_Calc_Figure_03_03_004.jpg\" alt=\"This figure has two parts labeled a and b. Figure a shows the graph of f(x) = x3 + 3x2 + x. Figure b shows the same graph but this time with two boxes on it. The first box appears along the left-hand side of the graph straddling the x-axis roughly parallel to f(x). The second box appears a little higher, also roughly parallel to f(x), with its front corner located at (\u22121.9, 2.8). Note that this corner is roughly in line with the direct path of the track before it started to turn.\" width=\"860\" height=\"462\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. (a) One section of the racetrack can be modeled by the function [latex]f(x)=x^3+3x^2+x[\/latex]. (b) The front corner of the grandstand is located at [latex](-1.9,2.8)[\/latex].<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<ol id=\"fs-id1169736655867\">\n<li>Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the [latex](x,y)[\/latex] coordinates of this point near the turn.<\/li>\n<li>Find the equation of the tangent line to the curve at this point.<\/li>\n<li>To determine whether the spectators are in danger in this scenario, find the [latex]x[\/latex]-coordinate of the point where the tangent line crosses the line [latex]y=2.8[\/latex]. Is this point safely to the right of the grandstand? Or are the spectators in danger?<\/li>\n<li>What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point [latex](-2.5,0.625)[\/latex]. What is the slope of the tangent line at this point?<\/li>\n<li>If a driver loses control as described in part 4, are the spectators safe?<\/li>\n<li>Should you proceed with the current design for the grandstand, or should the grandstands be moved?<\/li>\n<\/ol>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-341\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.3 Differentiation Rules. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":14,"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\":\"3.3 Differentiation Rules\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-341","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/341\/revisions"}],"predecessor-version":[{"id":4808,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/341\/revisions\/4808"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}