{"id":328,"date":"2021-02-04T01:11:45","date_gmt":"2021-02-04T01:11:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=328"},"modified":"2022-03-16T05:25:18","modified_gmt":"2022-03-16T05:25:18","slug":"derivative-basics","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/derivative-basics\/","title":{"raw":"Derivative Basics","rendered":"Derivative Basics"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Recognize the meaning of the tangent to a curve at a point<\/li>\r\n \t<li>Calculate the slope of a tangent line<\/li>\r\n \t<li>Identify the derivative as the limit of a difference quotient<\/li>\r\n \t<li>Calculate the derivative of a given function at a point<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169739218205\" class=\"bc-section section\">\r\n<h2>Tangent Lines<\/h2>\r\n<p id=\"fs-id1169739302308\">We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point [latex](a,f(a))[\/latex] to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of [latex]x[\/latex] near [latex]a[\/latex] and drawing a line through the points [latex](a,f(a))[\/latex] and [latex](x,f(x))[\/latex], as shown in Figure 2. The slope of this line is given by an equation in the form of a <strong>difference quotient<\/strong>:<\/p>\r\n\r\n<div id=\"fs-id1169738930229\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m_{\\sec}=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738929481\">We can also calculate the slope of a secant line to a function at a value [latex]a[\/latex] by using this equation and replacing [latex]x[\/latex] with [latex]a+h[\/latex], where [latex]h[\/latex] is a value close to [latex]0[\/latex]. We can then calculate the slope of the line through the points [latex](a,f(a))[\/latex] and [latex](a+h,f(a+h))[\/latex]. In this case, we find the secant line has a slope given by the following difference quotient with increment [latex]h[\/latex]:<\/p>\r\n\r\n<div id=\"fs-id1169738985511\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m_{\\sec}=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<div id=\"fs-id1169739095772\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738924962\">Let [latex]f[\/latex] be a function defined on an interval [latex]I[\/latex] containing [latex]a[\/latex]. If [latex]x\\ne a[\/latex] is in [latex]I[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1169738850293\" class=\"equation\" style=\"text-align: center;\">[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738879993\">is a <strong>difference quotient<\/strong>.<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1169739195558\">Also, if [latex]h\\ne 0[\/latex] is chosen so that [latex]a+h[\/latex] is in [latex]I[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1169739097936\" class=\"equation\" style=\"text-align: center;\">[latex]Q=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738913121\">is a difference quotient with increment [latex]h[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1169739040482\"><a href=\"https:\/\/www.geogebra.org\/m\/MeMdCUEm\" target=\"_blank\" rel=\"noopener\">View several Java applets on the development of the derivative.<\/a><\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1169738819917\">These two expressions for calculating the slope of a secant line are illustrated in Figure 2. We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"875\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205147\/CNX_Calc_Figure_03_01_002.jpg\" alt=\"This figure consists of two graphs labeled a and b. Figure a shows the Cartesian coordinate plane with 0, a, and x marked on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)) and (x, f(x)). There is also a straight line that crosses these two points (a, f(a)) and (x, f(x)). At the bottom of the graph, the equation msec = (f(x) - f(a))\/(x - a) is given. Figure b shows a similar graph, but this time a + h is marked on the x-axis instead of x. Consequently, the curve labeled y = f(x) passes through (a, f(a)) and (a + h, f(a + h)) as does the straight line. At the bottom of the graph, the equation msec = (f(a + h) - f(a))\/h is given.\" width=\"875\" height=\"442\" \/> Figure 2. We can calculate the slope of a secant line in either of two ways.[\/caption]\r\n<p id=\"fs-id1169738909226\">In Figure 3(a) we see that, as the values of [latex]x[\/latex] approach [latex]a[\/latex], the slopes of the secant lines provide better estimates of the rate of change of the function at [latex]a[\/latex]. Furthermore, the secant lines themselves approach the tangent line to the function at [latex]a[\/latex], which represents the limit of the secant lines. Similarly, Figure 3(b) shows that as the values of [latex]h[\/latex] get closer to 0, the secant lines also approach the tangent line. The slope of the tangent line at [latex]a[\/latex] is the rate of change of the function at [latex]a[\/latex], as shown in Figure 3(c).<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"956\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205151\/CNX_Calc_Figure_03_01_003.jpg\" alt=\"This figure consists of three graphs labeled a, b, and c. Figure a shows the Cartesian coordinate plane with 0, a, x2, and x1 marked in order on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)), (x2, f(x2)), and (x1, f(x1)). There are three straight lines: the first crosses (a, f(a)) and (x1, f(x1)); the second crosses (a, f(a)) and (x2, f(x2)); and the third only touches (a, f(a)), making it the tangent. At the bottom of the graph, the equation mtan = limx \u2192 a (f(x) - f(a))\/(x - a) is given. Figure b shows a similar graph, but this time a + h2 and a + h1 are marked on the x-axis instead of x2 and x1. Consequently, the curve labeled y = f(x) passes through (a, f(a)), (a + h2, f(a + h2)), and (a + h1, f(a + h1)) and the straight lines similarly cross the graph as in Figure a. At the bottom of the graph, the equation mtan = limh \u2192 0 (f(a + h) - f(a))\/h is given. Figure c shows only the curve labeled y = f(x) and its tangent at point (a, f(a)).\" width=\"956\" height=\"368\" \/> Figure 3. The secant lines approach the tangent line (shown in green) as the second point approaches the first.[\/caption]\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1169738875428\"><a href=\"https:\/\/demonstrations.wolfram.com\/DifferentiationMicroscope\/\" target=\"_blank\" rel=\"noopener\">You can use this site to explore graphs to see if they have a tangent line at a point.<\/a><\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1169738949813\">In Figure 4, we show the graph of [latex]f(x)=\\sqrt{x}[\/latex] and its tangent line at [latex](1,1)[\/latex] in a series of tighter intervals about [latex]x=1[\/latex]. As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of [latex]x[\/latex] close to 1. In fact, the graph of [latex]f(x)[\/latex] itself appears to be locally linear in the immediate vicinity of [latex]x=1[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"831\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205154\/CNX_Calc_Figure_03_01_008.jpg\" alt=\"This figure consists of four graphs labeled a, b, c, and d. Figure a shows the graphs of the square root of x and the equation y = (x + 1)\/2 with the x-axis going from 0 to 4 and the y-axis going from 0 to 2.5. The graphs of these two functions look very close near 1; there is a box around where these graphs look close. Figure b shows a close up of these same two functions in the area of the box from Figure a, specifically x going from 0 to 2 and y going from 0 to 1.4. Figure c is the same graph as Figure b, but this one has a box from 0 to 1.1 in the x coordinate and 0.8 and 1 on the y coordinate. There is an arrow indicating that this is blown up in Figure d. Figure d shows a very close picture of the box from Figure c, and the two functions appear to be touching for almost the entire length of the graph.\" width=\"831\" height=\"721\" \/> Figure 4. For values of [latex]x[\/latex] close to 1, the graph of [latex]f(x)=\\sqrt{x}[\/latex] and its tangent line appear to coincide.[\/caption]\r\n<p id=\"fs-id1169738971600\">Formally we may define the tangent line to the graph of a function as follows.<\/p>\r\n\r\n<div id=\"fs-id1169739274783\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738962299\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The <em>tangent line<\/em> to [latex]f(x)[\/latex] at [latex]a[\/latex] is the line passing through the point [latex](a,f(a))[\/latex] having slope<\/p>\r\n\r\n<div id=\"fs-id1169739226063\" class=\"equation\" style=\"text-align: center;\">[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738970954\">provided this limit exists.<\/p>\r\n<p id=\"fs-id1169738947329\">Equivalently, we may define the tangent line to [latex]f(x)[\/latex] at [latex]a[\/latex] to be the line passing through the point [latex](a,f(a))[\/latex] having slope<\/p>\r\n\r\n<div id=\"fs-id1169738970614\" class=\"equation\" style=\"text-align: center;\">[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\nprovided this limit exists.\r\n\r\n<\/div>\r\n<p id=\"fs-id1169739243045\">Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines. The definition requires you to recall two algebraic techniques and formulas: evaluating a function with variable inputs and using point-slope form to write an equation of a line.<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Evaluating a function with variable inputs<\/h3>\r\nFunctions can be evaluated for inputs that are variables or expressions. The process is the same as evaluating with a constant, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.\r\n<p style=\"text-align: center;\">Given [latex]f(x)=4x+1[\/latex], find [latex]f(h+1)[\/latex].<\/p>\r\nThis time, you substitute [latex](h+1)[\/latex] into the equation for <i>x.<\/i>\r\n<p style=\"text-align: center;\">[latex]f(h+1)=4(h+1)+1[\/latex]<\/p>\r\nUse the distributive property on the right side, and then combine like terms to simplify.\r\n<p style=\"text-align: center;\">[latex]f(h+1)=4h+4+1=4h+5[\/latex]<\/p>\r\nGiven [latex]f(x)=4x+1[\/latex], [latex]f(h+1)=4h+5[\/latex].\r\n\r\nWatch this video for more:\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=_bi0B2zibOg&amp;feature=emb_imp_woyt\r\n\r\n[reveal-answer q=\"266836\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266836\"]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\/ExDetermineVariousFunctionOutputsForAQuadraticFunction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex: Determine Various Function Outputs for a Quadratic Function\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: writing an equation of a line using point-slope form<\/h3>\r\n<strong>Point-slope form<\/strong> of a linear equation takes the form\r\n<p style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/p>\r\nwhere [latex]m[\/latex]\u00a0is the slope and [latex]{x}_{1 }\\text{ and } {y}_{1}[\/latex]\u00a0are the [latex]x\\text{ and }y[\/latex]\u00a0coordinates of a specific point through which the line passes.\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739218205\" class=\"bc-section section\">\r\n<div id=\"fs-id1169739298611\" class=\"textbook exercises\">\r\n<h3>Example: Finding a Tangent Line<\/h3>\r\n<p id=\"fs-id1169739030550\">Find the equation of the line tangent to the graph of [latex]f(x)=x^2[\/latex] at [latex]x=3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738899144\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738899144\"]\r\n<p id=\"fs-id1169738899144\">First find the slope of the tangent line. In this example, use the first definition above.<\/p>\r\n\r\n<div id=\"fs-id1169738970130\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} &amp; =\\underset{x\\to 3}{\\lim}\\frac{f(x)-f(3)}{x-3} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3} &amp; &amp; &amp; \\text{Substitute} \\, f(x)=x^2 \\, \\text{and} \\, f(3)=9. \\\\ &amp; =\\underset{x\\to 3}{\\lim}\\frac{(x-3)(x+3)}{x-3}=\\underset{x\\to 3}{\\lim}(x+3)=6 &amp; &amp; &amp; \\text{Factor the numerator to evaluate the limit.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739020694\">Next, find a point on the tangent line. Since the line is tangent to the graph of [latex]f(x)[\/latex] at [latex]x=3[\/latex], it passes through the point [latex](3,f(3))[\/latex]. We have [latex]f(3)=9[\/latex], so the tangent line passes through the point [latex](3,9)[\/latex].<\/p>\r\n<p id=\"fs-id1169738993948\">Using the point-slope equation of the line with the slope [latex]m=6[\/latex] and the point [latex](3,9)[\/latex], we obtain the line [latex]y-9=6(x-3)[\/latex]. Simplifying, we have [latex]y=6x-9[\/latex]. The graph of [latex]f(x)=x^2[\/latex] and its tangent line at [latex]x=3[\/latex] are shown in Figure 5.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205157\/CNX_Calc_Figure_03_01_005.jpg\" alt=\"This figure consists of the graphs of f(x) = x squared and y = 6x - 9. The graphs of these functions appear to touch at x = 3.\" width=\"487\" height=\"321\" \/> Figure 5. The tangent line to [latex]f(x)[\/latex] at [latex]x=3[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Finding a Tangent Line.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=418&amp;end=506&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\/3.1DefiningTheDerivative418to506_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.1 Defining the Derivative\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1169739223534\" class=\"textbook exercises\">\r\n<h3>Example: The Slope of a Tangent Line Revisited<\/h3>\r\n<p id=\"fs-id1169739031271\">Use the second definition to find the slope of the line tangent to the graph of [latex]f(x)=x^2[\/latex] at [latex]x=3[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169738885423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169738885423\"]\r\n<p id=\"fs-id1169738885423\">The steps are very similar to the previous example.<\/p>\r\n\r\n<div id=\"fs-id1169739043970\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} &amp; =\\underset{h\\to 0}{\\lim}\\frac{f(3+h)-f(3)}{h} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{(3+h)^2-9}{h} &amp; &amp; &amp; \\text{Substitute} \\, f(3+h)=(3+h)^2 \\, \\text{and} \\, f(3)=9. \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{9+6h+h^2-9}{h} &amp; &amp; &amp; \\text{Expand and simplify to evaluate the limit.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{h(6+h)}{h}=\\underset{h\\to 0}{\\lim}(6+h)=6 \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169738960593\">We obtained the same value for the slope of the tangent line by using the other definition, demonstrating that the formulas can be interchanged.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169738966727\" class=\"textbook exercises\">\r\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\r\n<p id=\"fs-id1169738935464\">Find the equation of the line tangent to the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] at [latex]x=2[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739001198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739001198\"]\r\n<p id=\"fs-id1169739001198\">We can use the first definition from before, but as we have seen, the results are the same if we use the other definition.<\/p>\r\n\r\n<div id=\"fs-id1169738961345\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} &amp; =\\underset{x\\to 2}{\\lim}\\frac{f(x)-f(2)}{x-2} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{\\frac{1}{x}-\\frac{1}{2}}{x-2} &amp; &amp; &amp; \\text{Substitute} \\, f(x)=\\frac{1}{x} \\, \\text{and} \\, f(2)=\\frac{1}{2}. \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{\\frac{1}{x}-\\frac{1}{2}}{x-2} \\cdot \\frac{2x}{2x} &amp; &amp; &amp; \\begin{array}{l}\\text{Multiply numerator and denominator by} \\, 2x \\, \\text{to} \\\\ \\text{simplify fractions.} \\end{array} \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{(2-x)}{(x-2)(2x)} &amp; &amp; &amp; \\text{Simplify.} \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{-1}{2x} &amp; &amp; &amp; \\text{Simplify using} \\, \\frac{2-x}{x-2}=-1, \\, \\text{for} \\, x\\ne 2. \\\\ &amp; =-\\frac{1}{4} &amp; &amp; &amp; \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1169739270491\">We now know that the slope of the tangent line is [latex]-\\frac{1}{4}[\/latex]. To find the equation of the tangent line, we also need a point on the line. We know that [latex]f(2)=\\frac{1}{2}[\/latex]. Since the tangent line passes through the point [latex](2,\\frac{1}{2})[\/latex] we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation [latex]y=-\\frac{1}{4}x+1[\/latex]. The graphs of [latex]f(x)=\\frac{1}{x}[\/latex] and [latex]y=-\\frac{1}{4}x+1[\/latex] are shown in Figure 6.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11205200\/CNX_Calc_Figure_03_01_006.jpg\" alt=\"This figure consists of the graphs of f(x) = 1\/x and y = -x\/4 + 1. The part of the graph f(x) = 1\/x in the first quadrant appears to touch the other function\u2019s graph at x = 2.\" width=\"487\" height=\"321\" \/> Figure 6. The line is tangent to [latex]f(x)[\/latex] at [latex]x=2[\/latex][\/caption][\/hidden-answer]<\/div>\r\n<div id=\"fs-id1169739038138\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739231715\">Find the slope of the line tangent to the graph of [latex]f(x)=\\sqrt{x}[\/latex] at [latex]x=4[\/latex].<\/p>\r\n[reveal-answer q=\"3776221\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"3776221\"]\r\n<p id=\"fs-id1169738973750\">Use either definition. Multiply the numerator and the denominator by a conjugate.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739236594\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739236594\"]\r\n<p id=\"fs-id1169739236594\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=738&amp;end=892&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[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\/3.1DefiningTheDerivative738to892_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.1 Defining the Derivative\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]204652[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>The Derivative of a Function at a Point<\/h2>\r\n<p id=\"fs-id1169739269344\">The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the<strong> derivative<\/strong>. The process of finding a derivative is called <strong>differentiation<\/strong>.<\/p>\r\n\r\n<div id=\"fs-id1169739269799\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1169738859079\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\r\n\r\n<div id=\"fs-id1169739179144\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169739032184\">provided this limit exists.<\/p>\r\n<p id=\"fs-id1169739032188\">Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\r\n\r\n<div id=\"fs-id1169739188551\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\r\nprovided this limit exists.\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739198956\" class=\"textbook exercises\">\r\n<h3>Example: Estimating a Derivative<\/h3>\r\n<p id=\"fs-id1169739204510\">For [latex]f(x)=x^2[\/latex], use a table to estimate [latex]f^{\\prime}(3)[\/latex] using the first definition above.<\/p>\r\n[reveal-answer q=\"fs-id1169739033831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739033831\"]\r\n<p id=\"fs-id1169739033831\">Create a table using values of [latex]x[\/latex] just below 3 and just above 3.<\/p>\r\n\r\n<table id=\"fs-id1169739301095\" class=\"unnumbered\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column header is x and the second column is (x2 \u2212 9)\/(x \u2212 3). Under the first column are the values 2.9, 2.99, 2.999, 3.001, 3.01, and 3.1. Under the second column are the values 5.9, 5.99, 5.999, 6.001, 6.01, and 6.1.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{x^2-9}{x-3}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2.9<\/td>\r\n<td>5.9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.99<\/td>\r\n<td>5.99<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.999<\/td>\r\n<td>5.999<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.001<\/td>\r\n<td>6.001<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.01<\/td>\r\n<td>6.01<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.1<\/td>\r\n<td>6.1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169739032830\">After examining the table, we see that a good estimate is [latex]f^{\\prime}(3)=6[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739304299\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739298320\">For [latex]f(x)=x^2[\/latex], use a table to estimate [latex]f^{\\prime}(3)[\/latex] using the second definition.<\/p>\r\n[reveal-answer q=\"8446220\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"8446220\"]\r\n<p id=\"fs-id1169739027513\">Evaluate [latex]\\frac{(x+h)^2-x^2}{h}[\/latex] at [latex]h=-0.1,-0.01,-0.001,0.001,0.01,0.1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739190379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739190379\"]\r\n<p id=\"fs-id1169739190379\">6<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739044032\" class=\"textbook exercises\">\r\n<h3>Example: Finding a Derivative<\/h3>\r\n<p id=\"fs-id1169739001692\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the first definition.<\/p>\r\n[reveal-answer q=\"fs-id1169739104712\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739104712\"]\r\n<p id=\"fs-id1169739104712\">Substitute the given function and value directly into the equation.<\/p>\r\n\r\n<div id=\"fs-id1169739305050\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(x)&amp; =\\underset{x\\to 2}{\\lim}\\frac{f(x)-f(2)}{x-2} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{(3x^2-4x+1)-5}{x-2} &amp; &amp; &amp; \\text{Substitute} \\, f(x)=3x^2-4x+1 \\, \\text{and} \\, f(2)=5. \\\\ &amp; =\\underset{x\\to 2}{\\lim}\\frac{(x-2)(3x+2)}{x-2} &amp; &amp; &amp; \\text{Simplify and factor the numerator.} \\\\ &amp; =\\underset{x\\to 2}{\\lim}(3x+2) &amp; &amp; &amp; \\text{Cancel the common factor.} \\\\ &amp; =8 &amp; &amp; &amp; \\text{Evaluate the limit.} \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1169739093789\" class=\"textbook exercises\">\r\n<h3>Example: Revisiting the Derivative<\/h3>\r\n<p id=\"fs-id1169739025974\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the second definition.<\/p>\r\n[reveal-answer q=\"fs-id1169739270484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739270484\"]\r\n<p id=\"fs-id1169739270484\">Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in the previous example.<\/p>\r\n\r\n<div id=\"fs-id1169736614162\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(2) &amp; =\\underset{h\\to 0}{\\lim}\\frac{f(2+h)-f(2)}{h} &amp; &amp; &amp; \\text{Apply the definition.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{(3(2+h)^2-4(2+h)+1)-5}{h} &amp; &amp; &amp; \\begin{array}{l}\\text{Substitute} \\, f(2)=5 \\, \\text{and} \\\\ f(2+h)=3(2+h)^2-4(2+h)+1. \\end{array} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{3h^2+8h}{h} &amp; &amp; &amp; \\text{Simplify the numerator.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}\\frac{h(3h+8)}{h} &amp; &amp; &amp; \\text{Factor the numerator.} \\\\ &amp; =\\underset{h\\to 0}{\\lim}(3h+8) &amp; &amp; &amp; \\text{Cancel the common factor.} \\\\ &amp; =8 &amp; &amp; &amp; \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169738850760\">The results are the same whether we use the first or second definition.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1169739236870\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1169739269938\">For [latex]f(x)=x^2+3x+2[\/latex], find [latex]f^{\\prime}(1)[\/latex].<\/p>\r\n[reveal-answer q=\"708365\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"708365\"]\r\n<p id=\"fs-id1169736618844\">Use either the first definition, the second, or try both. Use either the example for finding the equation of a tangent line or the example for estimating a derivative as a guide.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169739293550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739293550\"]\r\n<p id=\"fs-id1169739293550\">[latex]f^{\\prime}(1)=5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=1434&amp;end=1537&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266835\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266835\"]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\/3.1DefiningTheDerivative1434to1537_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.1 Defining the Derivative\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]162456[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Recognize the meaning of the tangent to a curve at a point<\/li>\n<li>Calculate the slope of a tangent line<\/li>\n<li>Identify the derivative as the limit of a difference quotient<\/li>\n<li>Calculate the derivative of a given function at a point<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169739218205\" class=\"bc-section section\">\n<h2>Tangent Lines<\/h2>\n<p id=\"fs-id1169739302308\">We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point [latex](a,f(a))[\/latex] to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of [latex]x[\/latex] near [latex]a[\/latex] and drawing a line through the points [latex](a,f(a))[\/latex] and [latex](x,f(x))[\/latex], as shown in Figure 2. The slope of this line is given by an equation in the form of a <strong>difference quotient<\/strong>:<\/p>\n<div id=\"fs-id1169738930229\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m_{\\sec}=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738929481\">We can also calculate the slope of a secant line to a function at a value [latex]a[\/latex] by using this equation and replacing [latex]x[\/latex] with [latex]a+h[\/latex], where [latex]h[\/latex] is a value close to [latex]0[\/latex]. We can then calculate the slope of the line through the points [latex](a,f(a))[\/latex] and [latex](a+h,f(a+h))[\/latex]. In this case, we find the secant line has a slope given by the following difference quotient with increment [latex]h[\/latex]:<\/p>\n<div id=\"fs-id1169738985511\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]m_{\\sec}=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1169739095772\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1169738924962\">Let [latex]f[\/latex] be a function defined on an interval [latex]I[\/latex] containing [latex]a[\/latex]. If [latex]x\\ne a[\/latex] is in [latex]I[\/latex], then<\/p>\n<div id=\"fs-id1169738850293\" class=\"equation\" style=\"text-align: center;\">[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738879993\">is a <strong>difference quotient<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739195558\">Also, if [latex]h\\ne 0[\/latex] is chosen so that [latex]a+h[\/latex] is in [latex]I[\/latex], then<\/p>\n<div id=\"fs-id1169739097936\" class=\"equation\" style=\"text-align: center;\">[latex]Q=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738913121\">is a difference quotient with increment [latex]h[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1169739040482\"><a href=\"https:\/\/www.geogebra.org\/m\/MeMdCUEm\" target=\"_blank\" rel=\"noopener\">View several Java applets on the development of the derivative.<\/a><\/p>\n<\/div>\n<p id=\"fs-id1169738819917\">These two expressions for calculating the slope of a secant line are illustrated in Figure 2. We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation.<\/p>\n<div style=\"width: 885px\" 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\/11205147\/CNX_Calc_Figure_03_01_002.jpg\" alt=\"This figure consists of two graphs labeled a and b. Figure a shows the Cartesian coordinate plane with 0, a, and x marked on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)) and (x, f(x)). There is also a straight line that crosses these two points (a, f(a)) and (x, f(x)). At the bottom of the graph, the equation msec = (f(x) - f(a))\/(x - a) is given. Figure b shows a similar graph, but this time a + h is marked on the x-axis instead of x. Consequently, the curve labeled y = f(x) passes through (a, f(a)) and (a + h, f(a + h)) as does the straight line. At the bottom of the graph, the equation msec = (f(a + h) - f(a))\/h is given.\" width=\"875\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. We can calculate the slope of a secant line in either of two ways.<\/p>\n<\/div>\n<p id=\"fs-id1169738909226\">In Figure 3(a) we see that, as the values of [latex]x[\/latex] approach [latex]a[\/latex], the slopes of the secant lines provide better estimates of the rate of change of the function at [latex]a[\/latex]. Furthermore, the secant lines themselves approach the tangent line to the function at [latex]a[\/latex], which represents the limit of the secant lines. Similarly, Figure 3(b) shows that as the values of [latex]h[\/latex] get closer to 0, the secant lines also approach the tangent line. The slope of the tangent line at [latex]a[\/latex] is the rate of change of the function at [latex]a[\/latex], as shown in Figure 3(c).<\/p>\n<div style=\"width: 966px\" 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\/11205151\/CNX_Calc_Figure_03_01_003.jpg\" alt=\"This figure consists of three graphs labeled a, b, and c. Figure a shows the Cartesian coordinate plane with 0, a, x2, and x1 marked in order on the x-axis. There is a curve labeled y = f(x) with points marked (a, f(a)), (x2, f(x2)), and (x1, f(x1)). There are three straight lines: the first crosses (a, f(a)) and (x1, f(x1)); the second crosses (a, f(a)) and (x2, f(x2)); and the third only touches (a, f(a)), making it the tangent. At the bottom of the graph, the equation mtan = limx \u2192 a (f(x) - f(a))\/(x - a) is given. Figure b shows a similar graph, but this time a + h2 and a + h1 are marked on the x-axis instead of x2 and x1. Consequently, the curve labeled y = f(x) passes through (a, f(a)), (a + h2, f(a + h2)), and (a + h1, f(a + h1)) and the straight lines similarly cross the graph as in Figure a. At the bottom of the graph, the equation mtan = limh \u2192 0 (f(a + h) - f(a))\/h is given. Figure c shows only the curve labeled y = f(x) and its tangent at point (a, f(a)).\" width=\"956\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. The secant lines approach the tangent line (shown in green) as the second point approaches the first.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1169738875428\"><a href=\"https:\/\/demonstrations.wolfram.com\/DifferentiationMicroscope\/\" target=\"_blank\" rel=\"noopener\">You can use this site to explore graphs to see if they have a tangent line at a point.<\/a><\/p>\n<\/div>\n<p id=\"fs-id1169738949813\">In Figure 4, we show the graph of [latex]f(x)=\\sqrt{x}[\/latex] and its tangent line at [latex](1,1)[\/latex] in a series of tighter intervals about [latex]x=1[\/latex]. As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of [latex]x[\/latex] close to 1. In fact, the graph of [latex]f(x)[\/latex] itself appears to be locally linear in the immediate vicinity of [latex]x=1[\/latex].<\/p>\n<div style=\"width: 841px\" 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\/11205154\/CNX_Calc_Figure_03_01_008.jpg\" alt=\"This figure consists of four graphs labeled a, b, c, and d. Figure a shows the graphs of the square root of x and the equation y = (x + 1)\/2 with the x-axis going from 0 to 4 and the y-axis going from 0 to 2.5. The graphs of these two functions look very close near 1; there is a box around where these graphs look close. Figure b shows a close up of these same two functions in the area of the box from Figure a, specifically x going from 0 to 2 and y going from 0 to 1.4. Figure c is the same graph as Figure b, but this one has a box from 0 to 1.1 in the x coordinate and 0.8 and 1 on the y coordinate. There is an arrow indicating that this is blown up in Figure d. Figure d shows a very close picture of the box from Figure c, and the two functions appear to be touching for almost the entire length of the graph.\" width=\"831\" height=\"721\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. For values of [latex]x[\/latex] close to 1, the graph of [latex]f(x)=\\sqrt{x}[\/latex] and its tangent line appear to coincide.<\/p>\n<\/div>\n<p id=\"fs-id1169738971600\">Formally we may define the tangent line to the graph of a function as follows.<\/p>\n<div id=\"fs-id1169739274783\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1169738962299\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The <em>tangent line<\/em> to [latex]f(x)[\/latex] at [latex]a[\/latex] is the line passing through the point [latex](a,f(a))[\/latex] having slope<\/p>\n<div id=\"fs-id1169739226063\" class=\"equation\" style=\"text-align: center;\">[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738970954\">provided this limit exists.<\/p>\n<p id=\"fs-id1169738947329\">Equivalently, we may define the tangent line to [latex]f(x)[\/latex] at [latex]a[\/latex] to be the line passing through the point [latex](a,f(a))[\/latex] having slope<\/p>\n<div id=\"fs-id1169738970614\" class=\"equation\" style=\"text-align: center;\">[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n<p>provided this limit exists.<\/p>\n<\/div>\n<p id=\"fs-id1169739243045\">Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines. The definition requires you to recall two algebraic techniques and formulas: evaluating a function with variable inputs and using point-slope form to write an equation of a line.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Evaluating a function with variable inputs<\/h3>\n<p>Functions can be evaluated for inputs that are variables or expressions. The process is the same as evaluating with a constant, but the simplified answer will contain a variable. The following examples show how to evaluate a function for a variable input.<\/p>\n<p style=\"text-align: center;\">Given [latex]f(x)=4x+1[\/latex], find [latex]f(h+1)[\/latex].<\/p>\n<p>This time, you substitute [latex](h+1)[\/latex] into the equation for <i>x.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]f(h+1)=4(h+1)+1[\/latex]<\/p>\n<p>Use the distributive property on the right side, and then combine like terms to simplify.<\/p>\n<p style=\"text-align: center;\">[latex]f(h+1)=4h+4+1=4h+5[\/latex]<\/p>\n<p>Given [latex]f(x)=4x+1[\/latex], [latex]f(h+1)=4h+5[\/latex].<\/p>\n<p>Watch this video for more:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine Various Function Outputs for a Quadratic Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_bi0B2zibOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266836\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266836\" 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\/ExDetermineVariousFunctionOutputsForAQuadraticFunction_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex: Determine Various Function Outputs for a Quadratic Function&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: writing an equation of a line using point-slope form<\/h3>\n<p><strong>Point-slope form<\/strong> of a linear equation takes the form<\/p>\n<p style=\"text-align: center;\">[latex]y-{y}_{1}=m\\left(x-{x}_{1}\\right)[\/latex]<\/p>\n<p>where [latex]m[\/latex]\u00a0is the slope and [latex]{x}_{1 }\\text{ and } {y}_{1}[\/latex]\u00a0are the [latex]x\\text{ and }y[\/latex]\u00a0coordinates of a specific point through which the line passes.<\/p>\n<\/div>\n<div id=\"fs-id1169739218205\" class=\"bc-section section\">\n<div id=\"fs-id1169739298611\" class=\"textbook exercises\">\n<h3>Example: Finding a Tangent Line<\/h3>\n<p id=\"fs-id1169739030550\">Find the equation of the line tangent to the graph of [latex]f(x)=x^2[\/latex] at [latex]x=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738899144\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738899144\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738899144\">First find the slope of the tangent line. In this example, use the first definition above.<\/p>\n<div id=\"fs-id1169738970130\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} & =\\underset{x\\to 3}{\\lim}\\frac{f(x)-f(3)}{x-3} & & & \\text{Apply the definition.} \\\\ & =\\underset{x\\to 3}{\\lim}\\frac{x^2-9}{x-3} & & & \\text{Substitute} \\, f(x)=x^2 \\, \\text{and} \\, f(3)=9. \\\\ & =\\underset{x\\to 3}{\\lim}\\frac{(x-3)(x+3)}{x-3}=\\underset{x\\to 3}{\\lim}(x+3)=6 & & & \\text{Factor the numerator to evaluate the limit.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739020694\">Next, find a point on the tangent line. Since the line is tangent to the graph of [latex]f(x)[\/latex] at [latex]x=3[\/latex], it passes through the point [latex](3,f(3))[\/latex]. We have [latex]f(3)=9[\/latex], so the tangent line passes through the point [latex](3,9)[\/latex].<\/p>\n<p id=\"fs-id1169738993948\">Using the point-slope equation of the line with the slope [latex]m=6[\/latex] and the point [latex](3,9)[\/latex], we obtain the line [latex]y-9=6(x-3)[\/latex]. Simplifying, we have [latex]y=6x-9[\/latex]. The graph of [latex]f(x)=x^2[\/latex] and its tangent line at [latex]x=3[\/latex] are shown in Figure 5.<\/p>\n<div style=\"width: 497px\" 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\/11205157\/CNX_Calc_Figure_03_01_005.jpg\" alt=\"This figure consists of the graphs of f(x) = x squared and y = 6x - 9. The graphs of these functions appear to touch at x = 3.\" width=\"487\" height=\"321\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. The tangent line to [latex]f(x)[\/latex] at [latex]x=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Finding a Tangent Line.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=418&amp;end=506&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\/3.1DefiningTheDerivative418to506_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.1 Defining the Derivative&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1169739223534\" class=\"textbook exercises\">\n<h3>Example: The Slope of a Tangent Line Revisited<\/h3>\n<p id=\"fs-id1169739031271\">Use the second definition to find the slope of the line tangent to the graph of [latex]f(x)=x^2[\/latex] at [latex]x=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169738885423\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169738885423\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738885423\">The steps are very similar to the previous example.<\/p>\n<div id=\"fs-id1169739043970\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} & =\\underset{h\\to 0}{\\lim}\\frac{f(3+h)-f(3)}{h} & & & \\text{Apply the definition.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{(3+h)^2-9}{h} & & & \\text{Substitute} \\, f(3+h)=(3+h)^2 \\, \\text{and} \\, f(3)=9. \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{9+6h+h^2-9}{h} & & & \\text{Expand and simplify to evaluate the limit.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{h(6+h)}{h}=\\underset{h\\to 0}{\\lim}(6+h)=6 \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169738960593\">We obtained the same value for the slope of the tangent line by using the other definition, demonstrating that the formulas can be interchanged.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169738966727\" class=\"textbook exercises\">\n<h3>Example: Finding the Equation of a Tangent Line<\/h3>\n<p id=\"fs-id1169738935464\">Find the equation of the line tangent to the graph of [latex]f(x)=\\dfrac{1}{x}[\/latex] at [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739001198\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739001198\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739001198\">We can use the first definition from before, but as we have seen, the results are the same if we use the other definition.<\/p>\n<div id=\"fs-id1169738961345\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}m_{\\tan} & =\\underset{x\\to 2}{\\lim}\\frac{f(x)-f(2)}{x-2} & & & \\text{Apply the definition.} \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{\\frac{1}{x}-\\frac{1}{2}}{x-2} & & & \\text{Substitute} \\, f(x)=\\frac{1}{x} \\, \\text{and} \\, f(2)=\\frac{1}{2}. \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{\\frac{1}{x}-\\frac{1}{2}}{x-2} \\cdot \\frac{2x}{2x} & & & \\begin{array}{l}\\text{Multiply numerator and denominator by} \\, 2x \\, \\text{to} \\\\ \\text{simplify fractions.} \\end{array} \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{(2-x)}{(x-2)(2x)} & & & \\text{Simplify.} \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{-1}{2x} & & & \\text{Simplify using} \\, \\frac{2-x}{x-2}=-1, \\, \\text{for} \\, x\\ne 2. \\\\ & =-\\frac{1}{4} & & & \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1169739270491\">We now know that the slope of the tangent line is [latex]-\\frac{1}{4}[\/latex]. To find the equation of the tangent line, we also need a point on the line. We know that [latex]f(2)=\\frac{1}{2}[\/latex]. Since the tangent line passes through the point [latex](2,\\frac{1}{2})[\/latex] we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation [latex]y=-\\frac{1}{4}x+1[\/latex]. The graphs of [latex]f(x)=\\frac{1}{x}[\/latex] and [latex]y=-\\frac{1}{4}x+1[\/latex] are shown in Figure 6.<\/p>\n<div style=\"width: 497px\" 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\/11205200\/CNX_Calc_Figure_03_01_006.jpg\" alt=\"This figure consists of the graphs of f(x) = 1\/x and y = -x\/4 + 1. The part of the graph f(x) = 1\/x in the first quadrant appears to touch the other function\u2019s graph at x = 2.\" width=\"487\" height=\"321\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. The line is tangent to [latex]f(x)[\/latex] at [latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739038138\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739231715\">Find the slope of the line tangent to the graph of [latex]f(x)=\\sqrt{x}[\/latex] at [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q3776221\">Hint<\/span><\/p>\n<div id=\"q3776221\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169738973750\">Use either definition. Multiply the numerator and the denominator by a conjugate.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739236594\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739236594\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739236594\">[latex]\\dfrac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=738&amp;end=892&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\/3.1DefiningTheDerivative738to892_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.1 Defining the Derivative&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm204652\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=204652&theme=oea&iframe_resize_id=ohm204652&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>The Derivative of a Function at a Point<\/h2>\n<p id=\"fs-id1169739269344\">The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the<strong> derivative<\/strong>. The process of finding a derivative is called <strong>differentiation<\/strong>.<\/p>\n<div id=\"fs-id1169739269799\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1169738859079\">Let [latex]f(x)[\/latex] be a function defined in an open interval containing [latex]a[\/latex]. The derivative of the function [latex]f(x)[\/latex] at [latex]a[\/latex], denoted by [latex]f^{\\prime}(a)[\/latex], is defined by<\/p>\n<div id=\"fs-id1169739179144\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169739032184\">provided this limit exists.<\/p>\n<p id=\"fs-id1169739032188\">Alternatively, we may also define the derivative of [latex]f(x)[\/latex] at [latex]a[\/latex] as<\/p>\n<div id=\"fs-id1169739188551\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/div>\n<p>provided this limit exists.<\/p>\n<\/div>\n<div id=\"fs-id1169739198956\" class=\"textbook exercises\">\n<h3>Example: Estimating a Derivative<\/h3>\n<p id=\"fs-id1169739204510\">For [latex]f(x)=x^2[\/latex], use a table to estimate [latex]f^{\\prime}(3)[\/latex] using the first definition above.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739033831\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739033831\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739033831\">Create a table using values of [latex]x[\/latex] just below 3 and just above 3.<\/p>\n<table id=\"fs-id1169739301095\" class=\"unnumbered\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column header is x and the second column is (x2 \u2212 9)\/(x \u2212 3). Under the first column are the values 2.9, 2.99, 2.999, 3.001, 3.01, and 3.1. Under the second column are the values 5.9, 5.99, 5.999, 6.001, 6.01, and 6.1.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{x^2-9}{x-3}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2.9<\/td>\n<td>5.9<\/td>\n<\/tr>\n<tr>\n<td>2.99<\/td>\n<td>5.99<\/td>\n<\/tr>\n<tr>\n<td>2.999<\/td>\n<td>5.999<\/td>\n<\/tr>\n<tr>\n<td>3.001<\/td>\n<td>6.001<\/td>\n<\/tr>\n<tr>\n<td>3.01<\/td>\n<td>6.01<\/td>\n<\/tr>\n<tr>\n<td>3.1<\/td>\n<td>6.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169739032830\">After examining the table, we see that a good estimate is [latex]f^{\\prime}(3)=6[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739304299\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739298320\">For [latex]f(x)=x^2[\/latex], use a table to estimate [latex]f^{\\prime}(3)[\/latex] using the second definition.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q8446220\">Hint<\/span><\/p>\n<div id=\"q8446220\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739027513\">Evaluate [latex]\\frac{(x+h)^2-x^2}{h}[\/latex] at [latex]h=-0.1,-0.01,-0.001,0.001,0.01,0.1[\/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-id1169739190379\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739190379\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739190379\">6<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739044032\" class=\"textbook exercises\">\n<h3>Example: Finding a Derivative<\/h3>\n<p id=\"fs-id1169739001692\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the first definition.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739104712\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739104712\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739104712\">Substitute the given function and value directly into the equation.<\/p>\n<div id=\"fs-id1169739305050\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(x)& =\\underset{x\\to 2}{\\lim}\\frac{f(x)-f(2)}{x-2} & & & \\text{Apply the definition.} \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{(3x^2-4x+1)-5}{x-2} & & & \\text{Substitute} \\, f(x)=3x^2-4x+1 \\, \\text{and} \\, f(2)=5. \\\\ & =\\underset{x\\to 2}{\\lim}\\frac{(x-2)(3x+2)}{x-2} & & & \\text{Simplify and factor the numerator.} \\\\ & =\\underset{x\\to 2}{\\lim}(3x+2) & & & \\text{Cancel the common factor.} \\\\ & =8 & & & \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739093789\" class=\"textbook exercises\">\n<h3>Example: Revisiting the Derivative<\/h3>\n<p id=\"fs-id1169739025974\">For [latex]f(x)=3x^2-4x+1[\/latex], find [latex]f^{\\prime}(2)[\/latex] by using the second definition.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739270484\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739270484\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739270484\">Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in the previous example.<\/p>\n<div id=\"fs-id1169736614162\" class=\"equation unnumbered\">[latex]\\begin{array}{lllll}f^{\\prime}(2) & =\\underset{h\\to 0}{\\lim}\\frac{f(2+h)-f(2)}{h} & & & \\text{Apply the definition.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{(3(2+h)^2-4(2+h)+1)-5}{h} & & & \\begin{array}{l}\\text{Substitute} \\, f(2)=5 \\, \\text{and} \\\\ f(2+h)=3(2+h)^2-4(2+h)+1. \\end{array} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{3h^2+8h}{h} & & & \\text{Simplify the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}\\frac{h(3h+8)}{h} & & & \\text{Factor the numerator.} \\\\ & =\\underset{h\\to 0}{\\lim}(3h+8) & & & \\text{Cancel the common factor.} \\\\ & =8 & & & \\text{Evaluate the limit.} \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169738850760\">The results are the same whether we use the first or second definition.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169739236870\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1169739269938\">For [latex]f(x)=x^2+3x+2[\/latex], find [latex]f^{\\prime}(1)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q708365\">Hint<\/span><\/p>\n<div id=\"q708365\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169736618844\">Use either the first definition, the second, or try both. Use either the example for finding the equation of a tangent line or the example for estimating a derivative as a guide.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1169739293550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1169739293550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1169739293550\">[latex]f^{\\prime}(1)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/VnDnInldaMM?controls=0&amp;start=1434&amp;end=1537&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=\"q266835\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266835\" 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\/3.1DefiningTheDerivative1434to1537_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.1 Defining the Derivative&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm162456\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=162456&theme=oea&iframe_resize_id=ohm162456&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-328\">\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>3.1 Defining the Derivative. <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":3,"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\":\"3.1 Defining the Derivative\",\"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-328","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/328","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":35,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/328\/revisions"}],"predecessor-version":[{"id":4801,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/328\/revisions\/4801"}],"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\/328\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=328"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=328"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=328"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}