{"id":2842,"date":"2024-06-10T17:54:12","date_gmt":"2024-06-10T17:54:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2842"},"modified":"2025-02-20T19:16:26","modified_gmt":"2025-02-20T19:16:26","slug":"the-derivative-as-a-function-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-derivative-as-a-function-fresh-take\/","title":{"raw":"The Derivative as a Function: Fresh Take","rendered":"The Derivative as a Function: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the derivative of a function<\/li>\r\n\t<li>Draw the derivative's graph using the original function\u2019s graph<\/li>\r\n\t<li>Explain what it means for a function to be differentiable and how this is connected to being continuous<\/li>\r\n\t<li>Calculate derivatives beyond the first order<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Derivative Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Derivative Function:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Gives the derivative of a function at every point in its domain<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Defined as: [latex]f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Differentiability:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">A function is differentiable at a point if its derivative exists at that point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">A function is differentiable on an interval if it's differentiable at every point in that interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation for Derivatives:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x)[\/latex], [latex]y'[\/latex], [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d}{dx}(f(x))[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Leibniz Notation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Represents instantaneous rate of change<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169737819520\">Find the derivative of [latex]f(x)=x^2[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737789025\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737789025\"]<\/p>\r\n<p id=\"fs-id1169737789025\">[latex]f^{\\prime}(x)=2x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\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=349&amp;end=408&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\/3.2TheDerivativeAsAFunction349to408_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<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the derivative of [latex]f(x) = x^3 + 2x[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"255451\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"255451\"]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] \\begin{array}{rcl} f'(x) &amp;=&amp; \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{((x+h)^3 + 2(x+h)) - (x^3 + 2x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{(x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h) - (x^3 + 2x)}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} \\frac{3x^2h + 3xh^2 + h^3 + 2h}{h} \\\\ &amp;=&amp; \\lim_{h \\to 0} (3x^2 + 3xh + h^2 + 2) \\\\ &amp;=&amp; 3x^2 + 2 \\end{array} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f'(x) = 3x^2 + 2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/N2PpRnFqnqY?si=3lS8rhkWPC-8ZiIl\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2>Graphing a Derivative<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Graphical Relationship:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">The derivative [latex]f'(x)[\/latex] represents the slope of the tangent line to [latex]f(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Positive [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] is increasing<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Negative [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] is decreasing<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Zero [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] has a horizontal tangent<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Features:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Local maxima\/minima of [latex]f(x)[\/latex] correspond to zeros of [latex]f'(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inflection points of [latex]f(x)[\/latex] correspond to local extrema of [latex]f'(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical asymptotes of [latex]f(x)[\/latex] may correspond to infinite limits of [latex]f'(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738223964\">Sketch the graph of [latex]f(x)=x^2-4[\/latex]. On what interval is the graph of [latex]f^{\\prime}(x)[\/latex] above the [latex]x[\/latex]-axis?<\/p>\r\n<p>[reveal-answer q=\"3005289\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"3005289\"]<\/p>\r\n<p id=\"fs-id1169738101726\">The graph of [latex]f^{\\prime}(x)[\/latex] is positive where [latex]f(x)[\/latex] is increasing.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737951946\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737951946\"]<\/p>\r\n<p id=\"fs-id1169737951946\">[latex](0,+\\infty)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Given the graph of [latex]f(x) = x^3 - 3x^2 + 2[\/latex], sketch the graph of [latex]f'(x)[\/latex].<\/p>\r\n\r\n\r\n[caption id=\"attachment_3715\" align=\"aligncenter\" width=\"562\"]<img class=\"size-full wp-image-3715\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035.png\" alt=\"A graph of the function f(x) = x\u00b3 - 3x\u00b2 + 2 on a coordinate plane. The blue curve starts in the lower left quadrant, rises steeply, reaches a local maximum around x=0, dips to a local minimum, then rises steeply again in the upper right quadrant. The graph shows two turning points, creating an S-shaped curve characteristic of a cubic function.\" width=\"562\" height=\"613\" \/> Graph of the cubic function f(x) = x\u00b3 - 3x\u00b2 + 2.[\/caption]\r\n\r\n\r\n<p><br \/>\r\n[reveal-answer q=\"857827\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"857827\"]<\/p>\r\n<p>The key points are:<\/p>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(0) = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(2) = 0[\/latex] (local maximum of [latex]f(x)[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(1) = -3[\/latex] (inflection point of [latex]f(x)[\/latex])<\/li>\r\n<\/ul>\r\n<p>Analyzing the intervals:<\/p>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x) &lt; 0[\/latex] for [latex]0 &lt; x &lt; 2[\/latex] ([latex]f(x)[\/latex] decreasing)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x) &gt; 0[\/latex] for [latex]x &lt; 0[\/latex] and [latex]x &gt; 2[\/latex] ([latex]f(x)[\/latex] increasing)<\/li>\r\n<\/ul>\r\n<p>Our sketch of [latex]f'(x)[\/latex] should include:<\/p>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Parabola opening upward<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts at [latex]x = 0[\/latex] and [latex]x = 2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertex at [latex](1, -3)[\/latex]<\/li>\r\n<\/ul>\r\n\r\n\r\n[caption id=\"attachment_3716\" align=\"aligncenter\" width=\"312\"]<img class=\"size-full wp-image-3716\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359.png\" alt=\"A graph of the quadratic function f(x) = 3x\u00b2 - 6x on a coordinate plane. The blue parabola opens upward, with its vertex slightly below the x-axis and to the right of the y-axis. The curve intersects the x-axis at two points, creating a U-shape.\" width=\"312\" height=\"455\" \/> Graph of f'(x), the quadratic function f(x) = 3x\u00b2 - 6x.[\/caption]\r\n\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/dIY86WqvGjw?si=me-A9eEXneTHircV\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2>Derivatives and Continuity<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Differentiability implies Continuity:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If a function is differentiable at a point, it must be continuous there<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The converse is not true: continuity does not imply differentiability<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Reasons for Non-differentiability:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sharp corners (e.g., absolute value function at [latex]x = 0[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical tangents (e.g., cube root function at [latex]x = 0[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Oscillating behavior (e.g., [latex]x \\sin(\\frac{1}{x})[\/latex] near [latex]x = 0[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Conditions for Differentiability:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Function must be continuous at the point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Left-hand and right-hand derivatives must exist and be equal<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find values of [latex]a[\/latex] and [latex]b[\/latex] that make the following function both continuous and differentiable at [latex]x = 2[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] f(x) = \\begin{cases} ax^2 + bx &amp; \\text{if } x &lt; 2 \\\\ x + 4 &amp; \\text{if } x \\geq 2 \\end{cases} [\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"87580\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"87580\"]<\/p>\r\n<p>For continuity at [latex]x = 2[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]4a + 2b = 6[\/latex]<\/p>\r\n<p>For differentiability at [latex]x = 2[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]2ax + b = 1[\/latex] when [latex]x = 2[\/latex]<br \/>\r\n[latex]4a + b = 1[\/latex]<\/p>\r\n<p>Solve the system of equations:<\/p>\r\n<p style=\"text-align: center;\">[latex]4a + 2b = 6[\/latex]<br \/>\r\n[latex]4a + b = 1[\/latex]<\/p>\r\n<p>Subtracting the second equation from the first:<\/p>\r\n<p style=\"text-align: center;\">[latex]b = 5[\/latex]<\/p>\r\n<p>Substituting back:<\/p>\r\n<p style=\"text-align: center;\">[latex]4a + 5 = 1[\/latex]<br \/>\r\n[latex]4a = -4[\/latex]<br \/>\r\n[latex]a = -1[\/latex]<\/p>\r\n<p>Therefore, [latex]a = -1[\/latex] and [latex]b = 5[\/latex] make the function both continuous and differentiable at [latex]x = 2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Analyze the continuity and differentiability of the following function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] f(x) = \\begin{cases} x^2 &amp; \\text{if } x \\leq 0 \\\\ \\sqrt{x} &amp; \\text{if } x &gt; 0 \\end{cases} [\/latex]<\/p>\r\n\r\n\r\n[reveal-answer q=\"660041\"]Show Answer[\/reveal-answer] [hidden-answer a=\"660041\"]\r\n\r\n\r\n<p>Check continuity at [latex]x = 0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0^-} f(x) = \\lim_{x \\to 0^-} x^2 = 0[\/latex]<br \/>\r\n[latex]\\lim_{x \\to 0^+} f(x) = \\lim_{x \\to 0^+} \\sqrt{x} = 0[\/latex]<br \/>\r\n[latex]f(0) = 0^2 = 0[\/latex]<\/p>\r\n<p>The function is continuous at [latex]x = 0[\/latex].<\/p>\r\n<br \/>\r\nCheck differentiability at [latex]x = 0[\/latex]:\r\n\r\n\r\n<p><br \/>\r\nLeft-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{f(0+h) - f(0)}{h} = \\lim_{h \\to 0^-} \\frac{h^2 - 0}{h} = \\lim_{h \\to 0^-} h = 0[\/latex]<\/p>\r\n<p>Right-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{f(0+h) - f(0)}{h} = \\lim_{h \\to 0^+} \\frac{\\sqrt{h} - 0}{h} = \\lim_{h \\to 0^+} \\frac{1}{\\sqrt{h}} = \\infty[\/latex]<\/p>\r\n<p>The left-hand and right-hand derivatives are not equal, so [latex]f(x)[\/latex] is not differentiable at [latex]x = 0[\/latex].<\/p>\r\n<p><strong>Conclusion:<\/strong> The function is continuous everywhere but not differentiable at [latex]x = 0[\/latex].<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find all points where the following function is not differentiable:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = |x^2 - 4x + 3|[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"916586\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"916586\"]<\/p>\r\n<p>The absolute value function can create points of non-differentiability where the expression inside changes sign.<\/p>\r\n<p><br \/>\r\nSet [latex]x^2 - 4x + 3 = 0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex](x - 1)(x - 3) = 0[\/latex]<br \/>\r\n[latex]x = 1[\/latex] or [latex]x = 3[\/latex]<\/p>\r\n<p>Check differentiability at [latex]x = 1[\/latex]:<\/p>\r\n<p>Left-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{|((1+h)^2 - 4(1+h) + 3)| - 0}{h} = \\lim_{h \\to 0^-} \\frac{|-h^2 - 2h|}{h} = -2[\/latex]<\/p>\r\n<p>Right-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{|((1+h)^2 - 4(1+h) + 3)| - 0}{h} = \\lim_{h \\to 0^+} \\frac{|-h^2 - 2h|}{h} = 2[\/latex]<\/p>\r\n<p>The function is not differentiable at [latex]x = 1[\/latex].<\/p>\r\n<p>Check differentiability at [latex]x = 3[\/latex]:<\/p>\r\n<p>Left-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{|((3+h)^2 - 4(3+h) + 3)| - 0}{h} = \\lim_{h \\to 0^-} \\frac{|h^2 + 2h|}{h} = 2[\/latex]<\/p>\r\n<p>Right-hand derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{|((3+h)^2 - 4(3+h) + 3)| - 0}{h} = \\lim_{h \\to 0^+} \\frac{|h^2 + 2h|}{h} = 2[\/latex]<\/p>\r\n<p>The function is differentiable at [latex]x = 3[\/latex].<\/p>\r\n<p><strong>Conclusion<\/strong>: The function is not differentiable at [latex]x = 1[\/latex], but is differentiable at all other points.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/xuAiQOzIkWY?si=vdJ_AxF08eSv3qUI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2>Higher-Order Derivatives<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Higher-order derivatives are the result of repeatedly differentiating a function<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Second derivative: derivative of the first derivative<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Third derivative: derivative of the second derivative, and so on<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For [latex]y = f(x)[\/latex]:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Second derivative: [latex]f''(x)[\/latex], [latex]y''[\/latex], or [latex]\\frac{d^2y}{dx^2}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Third derivative: [latex]f'''(x)[\/latex], [latex]y'''[\/latex], or [latex]\\frac{d^3y}{dx^3}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]n[\/latex]<sup>th<\/sup> derivative: [latex]f^{(n)}(x)[\/latex], [latex]y^{(n)}[\/latex], or [latex]\\frac{d^ny}{dx^n}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Physical Interpretations:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">First derivative of position: velocity<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Second derivative of position: acceleration<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Higher derivatives: rates of change of acceleration, etc.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738187258\">Find [latex]f''(x)[\/latex] for [latex]f(x)=x^2[\/latex].<\/p>\r\n\r\n\r\n[reveal-answer q=\"191037\"]Show Answer[\/reveal-answer] [hidden-answer a=\"191037\"]\r\n\r\n\r\n<p id=\"fs-id1169738099362\">[latex]f''(x)=2[\/latex]<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to this example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/w9m4h4i47-M?controls=0&amp;start=1432&amp;end=1510&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>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. You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction1432to1510_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>.<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the first four derivatives of [latex]f(x) = x^4 - 2x^3 + 3x^2 - 4x + 5[\/latex].<\/p>\r\n\r\n\r\n[reveal-answer q=\"807760\"]Show Answer[\/reveal-answer] [hidden-answer a=\"807760\"]\r\n\r\n\r\n<p>First derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = 4x^3 - 6x^2 + 6x - 4[\/latex]<\/p>\r\n<p>Second derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]f''(x) = 12x^2 - 12x + 6[\/latex]<\/p>\r\n<p>Third derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'''(x) = 24x - 12[\/latex]<\/p>\r\n<p>Fourth derivative:<\/p>\r\n<p style=\"text-align: center;\">[latex]f^{(4)}(x) = 24[\/latex]<\/p>\r\n<p>All subsequent derivatives will be zero.<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1169738185153\">For [latex]s(t)=t^3[\/latex], find [latex]a(t)[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1169737935177\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1169737935177\"]<\/p>\r\n<p id=\"fs-id1169737935177\">[latex]a(t)=6t[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/s7rd9YPJrNc?si=vMU9I0H1n8LO34-r\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/zUDxgehxQqs?si=DOsx_2axhDv2xmIP\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the derivative of a function<\/li>\n<li>Draw the derivative&#8217;s graph using the original function\u2019s graph<\/li>\n<li>Explain what it means for a function to be differentiable and how this is connected to being continuous<\/li>\n<li>Calculate derivatives beyond the first order<\/li>\n<\/ul>\n<\/section>\n<h2>Derivative Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Derivative Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Gives the derivative of a function at every point in its domain<\/li>\n<li class=\"whitespace-normal break-words\">Defined as: [latex]f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Differentiability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A function is differentiable at a point if its derivative exists at that point<\/li>\n<li class=\"whitespace-normal break-words\">A function is differentiable on an interval if it&#8217;s differentiable at every point in that interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Notation for Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(x)[\/latex], [latex]y'[\/latex], [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d}{dx}(f(x))[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Leibniz Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Represents instantaneous rate of change<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169737819520\">Find the derivative of [latex]f(x)=x^2[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737789025\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737789025\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737789025\">[latex]f^{\\prime}(x)=2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\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=349&amp;end=408&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\/3.2TheDerivativeAsAFunction349to408_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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the derivative of [latex]f(x) = x^3 + 2x[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q255451\">Show Answer<\/button><\/p>\n<div id=\"q255451\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{array}{rcl} f'(x) &=& \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{((x+h)^3 + 2(x+h)) - (x^3 + 2x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{(x^3 + 3x^2h + 3xh^2 + h^3 + 2x + 2h) - (x^3 + 2x)}{h} \\\\ &=& \\lim_{h \\to 0} \\frac{3x^2h + 3xh^2 + h^3 + 2h}{h} \\\\ &=& \\lim_{h \\to 0} (3x^2 + 3xh + h^2 + 2) \\\\ &=& 3x^2 + 2 \\end{array}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, [latex]f'(x) = 3x^2 + 2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/N2PpRnFqnqY?si=3lS8rhkWPC-8ZiIl\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2>Graphing a Derivative<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Graphical Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The derivative [latex]f'(x)[\/latex] represents the slope of the tangent line to [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Positive [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] is increasing<\/li>\n<li class=\"whitespace-normal break-words\">Negative [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] is decreasing<\/li>\n<li class=\"whitespace-normal break-words\">Zero [latex]f'(x)[\/latex] indicates [latex]f(x)[\/latex] has a horizontal tangent<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Local maxima\/minima of [latex]f(x)[\/latex] correspond to zeros of [latex]f'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inflection points of [latex]f(x)[\/latex] correspond to local extrema of [latex]f'(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical asymptotes of [latex]f(x)[\/latex] may correspond to infinite limits of [latex]f'(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738223964\">Sketch the graph of [latex]f(x)=x^2-4[\/latex]. On what interval is the graph of [latex]f^{\\prime}(x)[\/latex] above the [latex]x[\/latex]-axis?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3005289\">Hint<\/button><\/p>\n<div id=\"q3005289\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738101726\">The graph of [latex]f^{\\prime}(x)[\/latex] is positive where [latex]f(x)[\/latex] is increasing.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737951946\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737951946\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737951946\">[latex](0,+\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Given the graph of [latex]f(x) = x^3 - 3x^2 + 2[\/latex], sketch the graph of [latex]f'(x)[\/latex].<\/p>\n<figure id=\"attachment_3715\" aria-describedby=\"caption-attachment-3715\" style=\"width: 562px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3715\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035.png\" alt=\"A graph of the function f(x) = x\u00b3 - 3x\u00b2 + 2 on a coordinate plane. The blue curve starts in the lower left quadrant, rises steeply, reaches a local maximum around x=0, dips to a local minimum, then rises steeply again in the upper right quadrant. The graph shows two turning points, creating an S-shaped curve characteristic of a cubic function.\" width=\"562\" height=\"613\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035.png 562w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035-275x300.png 275w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035-65x71.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035-225x245.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153111\/Screenshot-2024-07-03-113035-350x382.png 350w\" sizes=\"(max-width: 562px) 100vw, 562px\" \/><figcaption id=\"caption-attachment-3715\" class=\"wp-caption-text\">Graph of the cubic function f(x) = x\u00b3 &#8211; 3x\u00b2 + 2.<\/figcaption><\/figure>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q857827\">Show Answer<\/button><\/p>\n<div id=\"q857827\" class=\"hidden-answer\" style=\"display: none\">\n<p>The key points are:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(0) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f'(2) = 0[\/latex] (local maximum of [latex]f(x)[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f'(1) = -3[\/latex] (inflection point of [latex]f(x)[\/latex])<\/li>\n<\/ul>\n<p>Analyzing the intervals:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(x) < 0[\/latex] for [latex]0 < x < 2[\/latex] ([latex]f(x)[\/latex] decreasing)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f'(x) > 0[\/latex] for [latex]x < 0[\/latex] and [latex]x > 2[\/latex] ([latex]f(x)[\/latex] increasing)<\/li>\n<\/ul>\n<p>Our sketch of [latex]f'(x)[\/latex] should include:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Parabola opening upward<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-intercepts at [latex]x = 0[\/latex] and [latex]x = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertex at [latex](1, -3)[\/latex]<\/li>\n<\/ul>\n<figure id=\"attachment_3716\" aria-describedby=\"caption-attachment-3716\" style=\"width: 312px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3716\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359.png\" alt=\"A graph of the quadratic function f(x) = 3x\u00b2 - 6x on a coordinate plane. The blue parabola opens upward, with its vertex slightly below the x-axis and to the right of the y-axis. The curve intersects the x-axis at two points, creating a U-shape.\" width=\"312\" height=\"455\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359.png 312w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359-206x300.png 206w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359-65x95.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/03153426\/Screenshot-2024-07-03-113359-225x328.png 225w\" sizes=\"(max-width: 312px) 100vw, 312px\" \/><figcaption id=\"caption-attachment-3716\" class=\"wp-caption-text\">Graph of f'(x), the quadratic function f(x) = 3x\u00b2 &#8211; 6x.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/dIY86WqvGjw?si=me-A9eEXneTHircV\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2>Derivatives and Continuity<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Differentiability implies Continuity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If a function is differentiable at a point, it must be continuous there<\/li>\n<li class=\"whitespace-normal break-words\">The converse is not true: continuity does not imply differentiability<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reasons for Non-differentiability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sharp corners (e.g., absolute value function at [latex]x = 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Vertical tangents (e.g., cube root function at [latex]x = 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Oscillating behavior (e.g., [latex]x \\sin(\\frac{1}{x})[\/latex] near [latex]x = 0[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conditions for Differentiability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Function must be continuous at the point<\/li>\n<li class=\"whitespace-normal break-words\">Left-hand and right-hand derivatives must exist and be equal<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find values of [latex]a[\/latex] and [latex]b[\/latex] that make the following function both continuous and differentiable at [latex]x = 2[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\begin{cases} ax^2 + bx & \\text{if } x < 2 \\\\ x + 4 & \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q87580\">Show Answer<\/button><\/p>\n<div id=\"q87580\" class=\"hidden-answer\" style=\"display: none\">\n<p>For continuity at [latex]x = 2[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]4a + 2b = 6[\/latex]<\/p>\n<p>For differentiability at [latex]x = 2[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]2ax + b = 1[\/latex] when [latex]x = 2[\/latex]<br \/>\n[latex]4a + b = 1[\/latex]<\/p>\n<p>Solve the system of equations:<\/p>\n<p style=\"text-align: center;\">[latex]4a + 2b = 6[\/latex]<br \/>\n[latex]4a + b = 1[\/latex]<\/p>\n<p>Subtracting the second equation from the first:<\/p>\n<p style=\"text-align: center;\">[latex]b = 5[\/latex]<\/p>\n<p>Substituting back:<\/p>\n<p style=\"text-align: center;\">[latex]4a + 5 = 1[\/latex]<br \/>\n[latex]4a = -4[\/latex]<br \/>\n[latex]a = -1[\/latex]<\/p>\n<p>Therefore, [latex]a = -1[\/latex] and [latex]b = 5[\/latex] make the function both continuous and differentiable at [latex]x = 2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p class=\"whitespace-pre-wrap break-words\">Analyze the continuity and differentiability of the following function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\begin{cases} x^2 & \\text{if } x \\leq 0 \\\\ \\sqrt{x} & \\text{if } x > 0 \\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q660041\">Show Answer<\/button> <\/p>\n<div id=\"q660041\" class=\"hidden-answer\" style=\"display: none\">\n<p>Check continuity at [latex]x = 0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{x \\to 0^-} f(x) = \\lim_{x \\to 0^-} x^2 = 0[\/latex]<br \/>\n[latex]\\lim_{x \\to 0^+} f(x) = \\lim_{x \\to 0^+} \\sqrt{x} = 0[\/latex]<br \/>\n[latex]f(0) = 0^2 = 0[\/latex]<\/p>\n<p>The function is continuous at [latex]x = 0[\/latex].<\/p>\n<p>\nCheck differentiability at [latex]x = 0[\/latex]:<\/p>\n<p>\nLeft-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{f(0+h) - f(0)}{h} = \\lim_{h \\to 0^-} \\frac{h^2 - 0}{h} = \\lim_{h \\to 0^-} h = 0[\/latex]<\/p>\n<p>Right-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{f(0+h) - f(0)}{h} = \\lim_{h \\to 0^+} \\frac{\\sqrt{h} - 0}{h} = \\lim_{h \\to 0^+} \\frac{1}{\\sqrt{h}} = \\infty[\/latex]<\/p>\n<p>The left-hand and right-hand derivatives are not equal, so [latex]f(x)[\/latex] is not differentiable at [latex]x = 0[\/latex].<\/p>\n<p><strong>Conclusion:<\/strong> The function is continuous everywhere but not differentiable at [latex]x = 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p class=\"whitespace-pre-wrap break-words\">Find all points where the following function is not differentiable:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = |x^2 - 4x + 3|[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q916586\">Show Answer<\/button><\/p>\n<div id=\"q916586\" class=\"hidden-answer\" style=\"display: none\">\n<p>The absolute value function can create points of non-differentiability where the expression inside changes sign.<\/p>\n<p>\nSet [latex]x^2 - 4x + 3 = 0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex](x - 1)(x - 3) = 0[\/latex]<br \/>\n[latex]x = 1[\/latex] or [latex]x = 3[\/latex]<\/p>\n<p>Check differentiability at [latex]x = 1[\/latex]:<\/p>\n<p>Left-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{|((1+h)^2 - 4(1+h) + 3)| - 0}{h} = \\lim_{h \\to 0^-} \\frac{|-h^2 - 2h|}{h} = -2[\/latex]<\/p>\n<p>Right-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{|((1+h)^2 - 4(1+h) + 3)| - 0}{h} = \\lim_{h \\to 0^+} \\frac{|-h^2 - 2h|}{h} = 2[\/latex]<\/p>\n<p>The function is not differentiable at [latex]x = 1[\/latex].<\/p>\n<p>Check differentiability at [latex]x = 3[\/latex]:<\/p>\n<p>Left-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^-} \\frac{|((3+h)^2 - 4(3+h) + 3)| - 0}{h} = \\lim_{h \\to 0^-} \\frac{|h^2 + 2h|}{h} = 2[\/latex]<\/p>\n<p>Right-hand derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\lim_{h \\to 0^+} \\frac{|((3+h)^2 - 4(3+h) + 3)| - 0}{h} = \\lim_{h \\to 0^+} \\frac{|h^2 + 2h|}{h} = 2[\/latex]<\/p>\n<p>The function is differentiable at [latex]x = 3[\/latex].<\/p>\n<p><strong>Conclusion<\/strong>: The function is not differentiable at [latex]x = 1[\/latex], but is differentiable at all other points.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/xuAiQOzIkWY?si=vdJ_AxF08eSv3qUI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2>Higher-Order Derivatives<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Higher-order derivatives are the result of repeatedly differentiating a function<\/li>\n<li class=\"whitespace-normal break-words\">Second derivative: derivative of the first derivative<\/li>\n<li class=\"whitespace-normal break-words\">Third derivative: derivative of the second derivative, and so on<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]y = f(x)[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Second derivative: [latex]f''(x)[\/latex], [latex]y''[\/latex], or [latex]\\frac{d^2y}{dx^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Third derivative: [latex]f'''(x)[\/latex], [latex]y'''[\/latex], or [latex]\\frac{d^3y}{dx^3}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex]<sup>th<\/sup> derivative: [latex]f^{(n)}(x)[\/latex], [latex]y^{(n)}[\/latex], or [latex]\\frac{d^ny}{dx^n}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Physical Interpretations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">First derivative of position: velocity<\/li>\n<li class=\"whitespace-normal break-words\">Second derivative of position: acceleration<\/li>\n<li class=\"whitespace-normal break-words\">Higher derivatives: rates of change of acceleration, etc.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738187258\">Find [latex]f''(x)[\/latex] for [latex]f(x)=x^2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q191037\">Show Answer<\/button> <\/p>\n<div id=\"q191037\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738099362\">[latex]f''(x)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">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=1432&amp;end=1510&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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. You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/3.2TheDerivativeAsAFunction1432to1510_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>.<\/section>\n<section class=\"textbox example\">\n<p>Find the first four derivatives of [latex]f(x) = x^4 - 2x^3 + 3x^2 - 4x + 5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q807760\">Show Answer<\/button> <\/p>\n<div id=\"q807760\" class=\"hidden-answer\" style=\"display: none\">\n<p>First derivative:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = 4x^3 - 6x^2 + 6x - 4[\/latex]<\/p>\n<p>Second derivative:<\/p>\n<p style=\"text-align: center;\">[latex]f''(x) = 12x^2 - 12x + 6[\/latex]<\/p>\n<p>Third derivative:<\/p>\n<p style=\"text-align: center;\">[latex]f'''(x) = 24x - 12[\/latex]<\/p>\n<p>Fourth derivative:<\/p>\n<p style=\"text-align: center;\">[latex]f^{(4)}(x) = 24[\/latex]<\/p>\n<p>All subsequent derivatives will be zero.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1169738185153\">For [latex]s(t)=t^3[\/latex], find [latex]a(t)[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1169737935177\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1169737935177\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169737935177\">[latex]a(t)=6t[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/s7rd9YPJrNc?si=vMU9I0H1n8LO34-r\" title=\"YouTube video player\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/zUDxgehxQqs?si=DOsx_2axhDv2xmIP\" title=\"YouTube video player\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"fresh_take","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2842"}],"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\/15"}],"version-history":[{"count":22,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2842\/revisions"}],"predecessor-version":[{"id":4686,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2842\/revisions\/4686"}],"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\/2842\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2842"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2842"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2842"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2842"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}