{"id":3014,"date":"2025-08-15T20:09:02","date_gmt":"2025-08-15T20:09:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=3014"},"modified":"2025-08-15T20:29:40","modified_gmt":"2025-08-15T20:29:40","slug":"derivatives-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/derivatives-learn-it-4\/","title":{"raw":"Derivatives: Learn It 4","rendered":"Derivatives: Learn It 4"},"content":{"raw":"

Finding Points Where a Function\u2019s Derivative Does Not Exist<\/h2>\r\nTo understand where a function\u2019s derivative does not exist, we need to recall what normally happens when a function [latex]f\\left(x\\right)[\/latex] has a derivative at [latex]x=a[\/latex] . Suppose we use a graphing utility to zoom in on [latex]x=a[\/latex] . If the function [latex]f\\left(x\\right)[\/latex] is differentiable<\/strong>, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity<\/em>.\r\n\r\nLook at the graph. The closer we zoom in on the point, the more linear the curve appears.\r\n\r\n\"Graph\r\n\r\nWe might presume the same thing would happen with any continuous function, but that is not so. The function [latex]f\\left(x\\right)=|x|[\/latex], for example, is continuous at [latex]x=0[\/latex], but not differentiable at [latex]x=0[\/latex]. As we zoom in close to 0, the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the function [latex]f\\left(x\\right)=|x|[\/latex], with x-axis from \u20130.1 to 0.1 and y-axis from \u20130.1 to 0.1.[\/caption]We zoom in closer by narrowing the range and continue to observe the same shape. This graph does not appear linear at [latex]x=0[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of the function [latex]f\\left(x\\right)=|x|[\/latex], with x-axis from \u20130.001 to 0.001 and y-axis from\u20140.001 to 0.001.[\/caption]What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function [latex]f\\left(x\\right)[\/latex] is not differentiable at [latex]x=a[\/latex].\r\n\r\nWe see the graph of\r\n

[latex]f\\left(x\\right)=\\begin{cases}x^{2}, \\hfill& x\\leq 2 \\\\ 8-x, \\hfill& x>2\\end{cases}[\/latex].<\/p>\r\nNotice that, as [latex]x[\/latex] approaches 2 from the left, the left-hand limit may be observed to be 4, while as [latex]x[\/latex] approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at [latex]x=2[\/latex].\r\n\r\n\"Graph\r\n\r\nThe graph of [latex]f\\left(x\\right)[\/latex] has a discontinuity at [latex]x=2[\/latex].\r\n\r\nWe see the graph of [latex]f\\left(x\\right)=|x|[\/latex]. We see that the graph has a corner point at [latex]x=0[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The graph of [latex]f\\left(x\\right)=|x|[\/latex] has a corner point at [latex]x=0[\/latex] .[\/caption]The graph of [latex]f\\left(x\\right)=|x|[\/latex] has a corner point at [latex]x=0[\/latex] .\r\n\r\nWe see that the graph of [latex]f\\left(x\\right)={x}^{\\frac{2}{3}}[\/latex] has a cusp at [latex]x=0[\/latex]. A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as [latex]x[\/latex] approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The graph of [latex]f\\left(x\\right)={x}^{\\frac{2}{3}}[\/latex] has a cusp at [latex]x=0[\/latex].[\/caption]We see that the graph of [latex]f\\left(x\\right)={x}^{\\frac{1}{3}}[\/latex] has a vertical tangent at [latex]x=0[\/latex]. Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The graph of [latex]f\\left(x\\right)={x}^{\\frac{1}{3}}[\/latex] has a vertical tangent<\/strong> at [latex]x=0[\/latex].[\/caption]

\r\n

differentiability<\/h3>\r\nA function [latex]f\\left(x\\right)[\/latex] is differentiable at [latex]x=a[\/latex] if the derivative exists at [latex]x=a[\/latex], which means that [latex]\\begin{align}{f}^{\\prime }\\left(a\\right)\\end{align}[\/latex] exists.\r\n\r\nThere are four cases for which a function [latex]f\\left(x\\right)[\/latex] is not differentiable at a point [latex]x=a[\/latex].\r\n
    \r\n \t
  1. When there is a discontinuity at [latex]x=a[\/latex].<\/li>\r\n \t
  2. When there is a corner point at [latex]x=a[\/latex].<\/li>\r\n \t
  3. When there is a cusp at [latex]x=a[\/latex].<\/li>\r\n \t
  4. Any other time when there is a vertical tangent at [latex]x=a[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Determine where the function is\r\n
      \r\n \t
    1. continuous<\/li>\r\n \t
    2. discontinuous<\/li>\r\n \t
    3. differentiable<\/li>\r\n \t
    4. not differentiable<\/li>\r\n<\/ol>\r\nAt the points where the graph is discontinuous or not differentiable, state why.\r\n\r\n\"Graph\r\n\r\n[reveal-answer q=\"912333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"912333\"]\r\n\r\nThe graph of [latex]f\\left(x\\right)[\/latex] is continuous on [latex]\\left(\\mathrm{-\\infty ,}-2\\right)\\cup \\left(-2, 1\\right)\\cup \\left(1,\\infty \\right)[\/latex]. The graph of [latex]f\\left(x\\right)[\/latex] has a removable discontinuity at [latex]x=-2[\/latex] and a jump discontinuity at [latex]x=1[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Three intervals where the function is continuous[\/caption]\r\n\r\nThe graph of is differentiable on [latex]\\left(\\mathrm{-\\infty ,}-2\\right)\\cup \\left(-2,-1\\right)\\cup \\left(-1,1\\right)\\cup \\left(1,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]. The graph of [latex]f\\left(x\\right)[\/latex] is not differentiable at [latex]x=-2[\/latex] because it is a point of discontinuity, at [latex]x=-1[\/latex] because of a sharp corner, at [latex]x=1[\/latex] because it is a point of discontinuity, and at [latex]x=2[\/latex] because of a sharp corner.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Five intervals where the function is differentiable[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      Determine where the function [latex]y=f\\left(x\\right)[\/latex] is continuous and differentiable from the graph.\"Graph[reveal-answer q=\"975733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"975733\"]\r\n\r\nThe graph of [latex]f[\/latex] is continuous on [latex]\\left(-\\infty ,1\\right)\\cup \\left(1,3\\right)\\cup \\left(3,\\infty \\right)[\/latex]. The graph of [latex]f[\/latex] is discontinuous at [latex]x=1[\/latex] and [latex]x=3[\/latex]. The graph of [latex]f[\/latex] is differentiable on [latex]\\left(-\\infty ,1\\right)\\cup \\left(1,3\\right)\\cup \\left(3,\\infty \\right)[\/latex]. The graph of [latex]f[\/latex] is not differentiable at [latex]x=1[\/latex] and [latex]x=3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

      Finding Points Where a Function\u2019s Derivative Does Not Exist<\/h2>\n

      To understand where a function\u2019s derivative does not exist, we need to recall what normally happens when a function [latex]f\\left(x\\right)[\/latex] has a derivative at [latex]x=a[\/latex] . Suppose we use a graphing utility to zoom in on [latex]x=a[\/latex] . If the function [latex]f\\left(x\\right)[\/latex] is differentiable<\/strong>, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity<\/em>.<\/p>\n

      Look at the graph. The closer we zoom in on the point, the more linear the curve appears.<\/p>\n

      \"Graph<\/p>\n

      We might presume the same thing would happen with any continuous function, but that is not so. The function [latex]f\\left(x\\right)=|x|[\/latex], for example, is continuous at [latex]x=0[\/latex], but not differentiable at [latex]x=0[\/latex]. As we zoom in close to 0, the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.<\/p>\n

      \"Graph
      Graph of the function [latex]f\\left(x\\right)=|x|[\/latex], with x-axis from \u20130.1 to 0.1 and y-axis from \u20130.1 to 0.1.<\/figcaption><\/figure>\n

      We zoom in closer by narrowing the range and continue to observe the same shape. This graph does not appear linear at [latex]x=0[\/latex].<\/p>\n

      \"Graph
      Graph of the function [latex]f\\left(x\\right)=|x|[\/latex], with x-axis from \u20130.001 to 0.001 and y-axis from\u20140.001 to 0.001.<\/figcaption><\/figure>\n

      What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function [latex]f\\left(x\\right)[\/latex] is not differentiable at [latex]x=a[\/latex].<\/p>\n

      We see the graph of<\/p>\n

      [latex]f\\left(x\\right)=\\begin{cases}x^{2}, \\hfill& x\\leq 2 \\\\ 8-x, \\hfill& x>2\\end{cases}[\/latex].<\/p>\n

      Notice that, as [latex]x[\/latex] approaches 2 from the left, the left-hand limit may be observed to be 4, while as [latex]x[\/latex] approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at [latex]x=2[\/latex].<\/p>\n

      \"Graph<\/p>\n

      The graph of [latex]f\\left(x\\right)[\/latex] has a discontinuity at [latex]x=2[\/latex].<\/p>\n

      We see the graph of [latex]f\\left(x\\right)=|x|[\/latex]. We see that the graph has a corner point at [latex]x=0[\/latex].<\/p>\n

      \"Graph
      The graph of [latex]f\\left(x\\right)=|x|[\/latex] has a corner point at [latex]x=0[\/latex] .<\/figcaption><\/figure>\n

      The graph of [latex]f\\left(x\\right)=|x|[\/latex] has a corner point at [latex]x=0[\/latex] .<\/p>\n

      We see that the graph of [latex]f\\left(x\\right)={x}^{\\frac{2}{3}}[\/latex] has a cusp at [latex]x=0[\/latex]. A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as [latex]x[\/latex] approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.<\/p>\n

      \"Graph
      The graph of [latex]f\\left(x\\right)={x}^{\\frac{2}{3}}[\/latex] has a cusp at [latex]x=0[\/latex].<\/figcaption><\/figure>\n

      We see that the graph of [latex]f\\left(x\\right)={x}^{\\frac{1}{3}}[\/latex] has a vertical tangent at [latex]x=0[\/latex]. Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.<\/p>\n

      \"Graph
      The graph of [latex]f\\left(x\\right)={x}^{\\frac{1}{3}}[\/latex] has a vertical tangent<\/strong> at [latex]x=0[\/latex].<\/figcaption><\/figure>\n
      \n

      differentiability<\/h3>\n

      A function [latex]f\\left(x\\right)[\/latex] is differentiable at [latex]x=a[\/latex] if the derivative exists at [latex]x=a[\/latex], which means that [latex]\\begin{align}{f}^{\\prime }\\left(a\\right)\\end{align}[\/latex] exists.<\/p>\n

      There are four cases for which a function [latex]f\\left(x\\right)[\/latex] is not differentiable at a point [latex]x=a[\/latex].<\/p>\n

        \n
      1. When there is a discontinuity at [latex]x=a[\/latex].<\/li>\n
      2. When there is a corner point at [latex]x=a[\/latex].<\/li>\n
      3. When there is a cusp at [latex]x=a[\/latex].<\/li>\n
      4. Any other time when there is a vertical tangent at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/section>\n
        Determine where the function is<\/p>\n
          \n
        1. continuous<\/li>\n
        2. discontinuous<\/li>\n
        3. differentiable<\/li>\n
        4. not differentiable<\/li>\n<\/ol>\n

          At the points where the graph is discontinuous or not differentiable, state why.<\/p>\n

          \"Graph<\/p>\n