{"id":333,"date":"2017-08-28T04:31:14","date_gmt":"2017-08-28T04:31:14","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=333"},"modified":"2018-03-02T05:08:38","modified_gmt":"2018-03-02T05:08:38","slug":"differential-calculus","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus\/","title":{"rendered":"Differential Calculus"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/limits-and-continuity\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus-ii\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<h1>Differential Calculus<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will cover the definition of a derivative and apply it to the evaluation of slopes and tangent lines of curves, instantaneous rates of changes of functions, relative maxima and minima, and methods to compute the derivative of several types of functions.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <strong><em>limit<\/em><\/strong> of a function is the value the function approaches as its variable approaches a number to the left and right of that number. The function does not need to be defined at that number, but must be defined for values slightly less and slightly more. Limits can be determined (if they exist) by plugging in the exact value and values on either side of the value itself, by graphing the function, or by mathematical manipulation, such as factoring.<\/li>\n<li>A <em><strong>one-sided limit<\/strong><\/em> approaches a value to one side of the variable but not the other. If a function is defined for numbers slightly higher than <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p1_html_m6a992457.gif\" name=\"graphics2\" align=\"BOTTOM\" width=\"13\" height=\"15\" border=\"0\"> but not for numbers slightly lower, the function has a right-hand limit, and we say that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image006.gif\" name=\"graphics3\" align=\"MIDDLE\" width=\"57\" height=\"23\" border=\"0\"> exists.<\/li>\n<li><em><strong>Infinite limits<\/strong><\/em> are limits that approach <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p1_html_65cbb539.gif\" name=\"graphics4\" align=\"BOTTOM\" width=\"25\" height=\"16\" border=\"0\"> as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image012.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> approaches <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image015.gif\" name=\"graphics6\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. This is represented graphically with a vertical asymptote at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image018.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"32\" height=\"7\" border=\"0\">.<\/li>\n<li>A <strong><em>limit at infinity<\/em><\/strong> is the value of a function as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image020.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> approaches <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p1_html_65cbb539.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"25\" height=\"16\" border=\"0\">. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image025.gif\" name=\"graphics10\" align=\"MIDDLE\" width=\"80\" height=\"20\" border=\"0\">, the limit at infinity is represented with a horizontal asymptote at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image028.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"35\" height=\"14\" border=\"0\">.<\/li>\n<li>A function is <em><strong>continuous<\/strong><\/em> over an interval if there are no breaks in the graph. In other words, for all points <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p1_html_m588f3dc7.gif\" name=\"graphics12\" align=\"BOTTOM\" width=\"37\" height=\"27\" border=\"0\"> in the interval (except the endpoints if the interval is open), <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image033.gif\" name=\"graphics13\" align=\"MIDDLE\" width=\"102\" height=\"20\" border=\"0\">, and terms both to the left and right of the equal sign exist.<\/li>\n<li>The <em><strong>Intermediate Value Theorem<\/strong><\/em> states that, if a function is continuous over a closed interval, every value of the function between the endpoints exists at least once, and there is a value of the variable within the open range that defines the function\u2019s value.<\/li>\n<\/ul>\n<h3>What is the derivative of a function?<\/h3>\n<p>A <strong><abbr rel=\"tooltip\" title=\"The derivative of a function  MathType is MathType.  MathType MathType can be interpreted as the slope of the tangent line of  MathType at MathType, or the instantaneous rate of change of  MathType at  MathType. In the context of higher order derivatives, the above definition applies to a first-order derivative.\">derivative<\/abbr><\/strong> is nothing more than a special type of limit. However, it has many important applications and interpretations. Substitute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image003.gif\" name=\"graphics3\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\"> for <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image006.gif\" name=\"graphics4\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> in our limit equation. We usually denote the derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image009.gif\" name=\"graphics5\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image012.gif\" name=\"graphics6\" align=\"ABSMIDDLE\" width=\"72\" height=\"34\" border=\"0\"> and define it as:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"190\" height=\"34\" border=\"0\">.<\/p>\n<p>Given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image018.gif\" name=\"graphics8\" align=\"ABSMIDDLE\" width=\"122\" height=\"17\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"64\" height=\"13\" border=\"0\"> are constants, compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image024.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\">.<\/p>\n<p>We will use the above limit to compute the derivative. First, we need to compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image027.gif\" name=\"graphics11\" align=\"ABSMIDDLE\" width=\"62\" height=\"14\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image030.gif\" name=\"graphics12\" align=\"BOTTOM\" width=\"495\" height=\"17\" border=\"0\"><\/p>\n<p>Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image033.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"485\" height=\"17\" border=\"0\">. Hence, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image036.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"217\" height=\"34\" border=\"0\">. Finally, we simply plug in values to find the derivative.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image039.gif\" name=\"graphics15\" align=\"ABSMIDDLE\" width=\"397\" height=\"34\" border=\"0\">. We will use this derivation for a couple of examples, so keep this solution handy.<\/p>\n<h3>What is the derivative of a constant?<\/h3>\n<p>Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image042.gif\" name=\"graphics16\" align=\"ABSMIDDLE\" width=\"53\" height=\"14\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image045.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"6\" height=\"7\" border=\"0\"> is a constant for all values of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image048.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. That means <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image051.gif\" name=\"graphics19\" align=\"ABSMIDDLE\" width=\"86\" height=\"14\" border=\"0\"> as well. Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image054.gif\" name=\"graphics20\" align=\"ABSMIDDLE\" width=\"342\" height=\"34\" border=\"0\">. So the derivative of a constant is zero.<\/p>\n<p>Compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image057.gif\" name=\"graphics21\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> for<\/p>\n<p>(A) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image060.gif\" name=\"graphics22\" align=\"ABSMIDDLE\" width=\"75\" height=\"14\" border=\"0\">; and <\/p>\n<p>(B) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image063.gif\" name=\"graphics23\" align=\"ABSMIDDLE\" width=\"78\" height=\"14\" border=\"0\">.<\/p>\n<p><em>Hint: <\/em>Remember the two limits for <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image066.gif\" name=\"graphics24\" align=\"BOTTOM\" width=\"27\" height=\"11\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image069.gif\" name=\"graphics25\" align=\"BOTTOM\" width=\"30\" height=\"7\" border=\"0\"> for small values of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image071.gif\" name=\"graphics26\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. Also, recall the following trigonometric identities:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image074.gif\" name=\"graphics27\" align=\"BOTTOM\" width=\"216\" height=\"38\" border=\"0\"><\/p>\n<p>(A) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image076.gif\" name=\"graphics28\" align=\"BOTTOM\" width=\"75\" height=\"14\" border=\"0\"><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p2_html_33dd6bdc.gif\" name=\"graphics29\" align=\"BOTTOM\" width=\"537\" height=\"133\" border=\"0\"><\/p>\n<p>(B) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image081.gif\" name=\"graphics30\" align=\"BOTTOM\" width=\"78\" height=\"14\" border=\"0\"><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p2_html_m3aedb682.gif\" name=\"graphics31\" align=\"BOTTOM\" width=\"561\" height=\"133\" border=\"0\"><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>\tThese two derivatives are significant to many applications in calculus and should be memorized.<\/p>\n<p>\tThe derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image087.gif\" name=\"graphics32\" align=\"BOTTOM\" width=\"28\" height=\"11\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image090.gif\" name=\"graphics33\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">.<\/p>\n<p>\tThe derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image092.gif\" name=\"graphics34\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p2_clip_image095.gif\" name=\"graphics35\" align=\"BOTTOM\" width=\"40\" height=\"11\" border=\"0\">.\n<\/div>\n<h3>What are the derivative and the slope of a tangent line?<\/h3>\n<p>If a function is in the form <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"68\" height=\"14\" border=\"0\">, recall that the graph crosses the <\/span><em><span style=\"text-decoration: none\">y<\/span><\/em><span style=\"text-decoration: none\">-axis at the <\/span><em><span style=\"text-decoration: none\">y<\/span><\/em><span style=\"text-decoration: none\">-intercept <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"30\" height=\"14\" border=\"0\"> and has a constant slope <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image009.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"11\" height=\"7\" border=\"0\">.<\/span><\/p>\n<p>The concept of slope is not limited to straight lines. Curves can have slopes as well, but the value of the slope of the curve <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image012.gif\" name=\"graphics6\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> changes as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> varies. As you may have guessed by now, the slope of the curve <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image017.gif\" name=\"graphics8\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> at any point <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image019.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> is defined by <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image022.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"35\" height=\"17\" border=\"0\">. (Actually, the slope is not always defined; at these points, the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image024.gif\" name=\"graphics11\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> is not defined.) Below we will show why.<\/p>\n<p>Consider the curve below and the line intersecting the curve at points <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image027.gif\" name=\"graphics12\" align=\"ABSMIDDLE\" width=\"62\" height=\"16\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image030.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"65\" height=\"16\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><strong><span style=\"text-decoration: none\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.1%20Art%20005.jpg\" name=\"graphics14\" alt=\"Intersecting Curves\" align=\"BOTTOM\" width=\"300\" height=\"287\" border=\"0\"><\/span><\/strong><\/p>\n<p>A line that intersects the curve at two points is known as a <abbr rel=\"tooltip\" title=\"any line intersecting a curve in two points\">secant line<\/abbr>. The slope of that line is given by:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image033.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"223\" height=\"40\" border=\"0\"><\/p>\n<p>Here, we define <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image036.gif\" name=\"graphics16\" align=\"ABSMIDDLE\" width=\"74\" height=\"16\" border=\"0\">. Therefore, we can rewrite the above equation as:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image039.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"254\" height=\"34\" border=\"0\"><\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.1%20Art%20006.jpg\" name=\"graphics18\" alt=\" Graph of a curve with two intersecting lines\" align=\"BOTTOM\" width=\"300\" height=\"287\" border=\"0\"><\/strong><\/p>\n<p>Let\u2019s re-examine the above graph with another line that intersects the curve at just one point. This new line is known as the <abbr rel=\"tooltip\" title=\"any line that intersects a curve in exactly one point\">tangent line<\/abbr> at the point of intersection, and describes the slope of the curve at that point. As you can see, it does not have the same slope as the secant line.<\/p>\n<p>Imagine shifting the secant line so that the points of intersection become closer and closer to the tangent point. The slope of the secant line is still described by the above equation, but the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image042.gif\" name=\"graphics19\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\"> gets smaller and smaller. In fact, the slope of the secant line and tangent line become identical as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image045.gif\" name=\"graphics20\" align=\"BOTTOM\" width=\"104\" height=\"14\" border=\"0\">. We cannot simply plug in <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image048.gif\" name=\"graphics21\" align=\"BOTTOM\" width=\"41\" height=\"11\" border=\"0\">, because it would result in a zero in the denominator. What we can do, however, is consider the <abbr rel=\"tooltip\" title=\"the value of the function as its variable approaches a number. We read  c as, \u201cthe limit of  c as x approaches  c is B.\u201d For every number in an open interval around a (except perhaps a itself), the limit  c exists if, for any  c, there exists  c such that c.\">limit<\/abbr>of the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image050.gif\" name=\"graphics22\" align=\"BOTTOM\" width=\"11\" height=\"7\" border=\"0\"> as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image052.gif\" name=\"graphics23\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\"> approaches zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image055.gif\" name=\"graphics24\" align=\"ABSMIDDLE\" width=\"196\" height=\"34\" border=\"0\">, which is precisely the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image057.gif\" name=\"graphics25\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\">.<\/p>\n<p>For example, given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image060.gif\" name=\"graphics26\" align=\"ABSMIDDLE\" width=\"116\" height=\"17\" border=\"0\">, what is the slope of the curve at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image063.gif\" name=\"graphics27\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">?<\/p>\n<p>The graph of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image065.gif\" name=\"graphics28\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> and its tangent line at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image067.gif\" name=\"graphics29\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\"> are depicted below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.1%20Art%20oo7.jpg\" name=\"graphics30\" alt=\"graph of f(x) and its tangent line\" align=\"BOTTOM\" width=\"300\" height=\"290\" border=\"0\"><\/p>\n<p>It appears that the tangent line intersects t<span style=\"text-decoration: none\">he <\/span><em><span style=\"text-decoration: none\">x<\/span><\/em><span style=\"text-decoration: none\">-axis at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image069.gif\" name=\"graphics31\" align=\"ABSMIDDLE\" width=\"53\" height=\"14\" border=\"0\"> and the <\/span><em><span style=\"text-decoration: none\">y<\/span><\/em><span style=\"text-decoration: none\">-axis at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image072.gif\" name=\"graphics32\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\">. From these points, it appears that the slope is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image075.gif\" name=\"graphics33\" align=\"ABSMIDDLE\" width=\"72\" height=\"34\" border=\"0\">. We can verify this by taking the derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image077.gif\" name=\"graphics34\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> and plugging in <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image079.gif\" name=\"graphics35\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">. We already calculated the derivative: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image082.gif\" name=\"graphics36\" align=\"ABSMIDDLE\" width=\"90\" height=\"14\" border=\"0\">. At <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image084.gif\" name=\"graphics37\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p3_clip_image087.gif\" name=\"graphics38\" align=\"ABSMIDDLE\" width=\"102\" height=\"14\" border=\"0\">.<\/span><\/p>\n<h3>How do we find instantaneous rate of change?<\/h3>\n<p>Now that we understand that the derivative of a function is equivalent to the slope of a function at a point (or to the entire function, if the function describes a line), it should be easy to equate it to a rate of change. Recall that the slope of a line is defined as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"46\" height=\"34\" border=\"0\">. In this form, the slope describes the rate of change in <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"9\" height=\"10\" border=\"0\"> with respect to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image009.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. In physics applications, we associate the wo<span style=\"text-decoration: none\">rds <\/span><em><span style=\"text-decoration: none\">rate of change <\/span><\/em><span style=\"text-decoration: none\">with time. In economics, <\/span><em><span style=\"text-decoration: none\">rate of change<\/span><\/em><span style=\"text-decoration: none\"> might be associated with the rate of change of cost (such as marginal cost). <\/span><\/p>\n<p>Let\u2019s start with a physics example. Suppose the position of a particle over time can be described by the equation <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image012.gif\" name=\"graphics6\" align=\"ABSMIDDLE\" width=\"108\" height=\"17\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image015.gif\" name=\"graphics7\" align=\"ABSMIDDLE\" width=\"22\" height=\"14\" border=\"0\"> is measured in meters and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image018.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"4\" height=\"9\" border=\"0\"> in seconds. At time <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">, the particle is at a position <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image024.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"58\" height=\"14\" border=\"0\"> (think of a race car starting <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"7\" height=\"11\" border=\"0\"> meters behind the starting line). Also note that, as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image029.gif\" name=\"graphics12\" align=\"BOTTOM\" width=\"4\" height=\"9\" border=\"0\"> increases, the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image031.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"22\" height=\"14\" border=\"0\"> not only grows, but grows at a greater and greater rate (we usually do not look at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image034.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"34\" height=\"14\" border=\"0\">.<\/p>\n<p><span style=\"text-decoration: none\">For example, what is the<\/span><em><span style=\"text-decoration: none\">average<\/span><\/em><span style=\"text-decoration: none\"> velocity of the particle from <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image036.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image039.gif\" name=\"graphics16\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> seconds? Recall from algebra that rate (velocity) = distance\/time. Between <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image041.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image044.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"28\" height=\"11\" border=\"0\"> seconds, the particle has moved from position <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image046.gif\" name=\"graphics19\" align=\"BOTTOM\" width=\"58\" height=\"14\" border=\"0\"> meters to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image049.gif\" name=\"graphics20\" align=\"ABSMIDDLE\" width=\"166\" height=\"14\" border=\"0\"> meters, for a total of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image052.gif\" name=\"graphics21\" align=\"BOTTOM\" width=\"14\" height=\"11\" border=\"0\"> m. This distance is covered over a period of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image055.gif\" name=\"graphics22\" align=\"BOTTOM\" width=\"6\" height=\"11\" border=\"0\"> seconds. Therefore, the average velocity is 26\/2 = 13 meters per second. <\/span><\/p>\n<h3><span style=\"text-decoration: none\">What is the <\/span><em><span style=\"text-decoration: none\">instantaneous<\/span><\/em><span style=\"text-decoration: none\"> velocity <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image058.gif\" name=\"graphics23\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image060.gif\" name=\"graphics24\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> seconds?<\/span><\/h3>\n<p>We see that the average velocity over a time <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image063.gif\" name=\"graphics25\" align=\"BOTTOM\" width=\"14\" height=\"11\" border=\"0\"> is described by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image066.gif\" name=\"graphics26\" align=\"ABSMIDDLE\" width=\"168\" height=\"34\" border=\"0\">. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image069.gif\" name=\"graphics27\" align=\"ABSMIDDLE\" width=\"34\" height=\"15\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image072.gif\" name=\"graphics28\" align=\"BOTTOM\" width=\"39\" height=\"11\" border=\"0\"> sec. What happens as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image074.gif\" name=\"graphics29\" align=\"BOTTOM\" width=\"14\" height=\"11\" border=\"0\"> gets smaller and smaller? We can then measure the small change in position over a small interval in time. The average velocity then becomes close to the instantaneous velocity around that time. The instantaneous velocity is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image077.gif\" name=\"graphics30\" align=\"ABSMIDDLE\" width=\"48\" height=\"17\" border=\"0\"> as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image079.gif\" name=\"graphics31\" align=\"BOTTOM\" width=\"14\" height=\"11\" border=\"0\"> approaches zero, or:<\/p>\n<p><span style=\"text-decoration: none\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image082.gif\" name=\"graphics32\" align=\"ABSMIDDLE\" width=\"141\" height=\"34\" border=\"0\">. Note that this is the formula for a derivative. Except we now use <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image085.gif\" name=\"graphics33\" align=\"BOTTOM\" width=\"6\" height=\"7\" border=\"0\"> instead of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image088.gif\" name=\"graphics34\" align=\"BOTTOM\" width=\"10\" height=\"14\" border=\"0\">, and the variable is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image091.gif\" name=\"graphics35\" align=\"BOTTOM\" width=\"4\" height=\"9\" border=\"0\"> instead of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image094.gif\" name=\"graphics36\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. We can therefore denote <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image097.gif\" name=\"graphics37\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> by the terms <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image100.gif\" name=\"graphics38\" align=\"ABSMIDDLE\" width=\"25\" height=\"14\" border=\"0\"> or <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image103.gif\" name=\"graphics39\" align=\"ABSMIDDLE\" width=\"16\" height=\"34\" border=\"0\">. Rather than expand the limit term, let\u2019s just borrow what we found in the first example. For the function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image106.gif\" name=\"graphics40\" align=\"ABSMIDDLE\" width=\"118\" height=\"17\" border=\"0\">, we found <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image109.gif\" name=\"graphics41\" align=\"ABSMIDDLE\" width=\"95\" height=\"14\" border=\"0\">. Just substitute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image112.gif\" name=\"graphics42\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image115.gif\" name=\"graphics43\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image118.gif\" name=\"graphics44\" align=\"BOTTOM\" width=\"39\" height=\"11\" border=\"0\"> (as well as changing <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image120.gif\" name=\"graphics45\" align=\"ABSMIDDLE\" width=\"10\" height=\"14\" border=\"0\"> to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image122.gif\" name=\"graphics46\" align=\"BOTTOM\" width=\"6\" height=\"7\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image124.gif\" name=\"graphics47\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image126.gif\" name=\"graphics48\" align=\"BOTTOM\" width=\"4\" height=\"9\" border=\"0\">). We find: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image129.gif\" name=\"graphics49\" align=\"ABSMIDDLE\" width=\"123\" height=\"14\" border=\"0\">. This equation describ<\/span>es the instantaneous velocity at all times. At <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image131.gif\" name=\"graphics50\" align=\"BOTTOM\" width=\"28\" height=\"11\" border=\"0\"> second, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image134.gif\" name=\"graphics51\" align=\"ABSMIDDLE\" width=\"58\" height=\"14\" border=\"0\">meters per second.<\/p>\n<h3>What is the acceleration of the particle?<\/h3>\n<p>The acceleration is defined as the change in velocity over time. Using similar reasoning to find the instantaneous velocity, the acceleration <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image137.gif\" name=\"graphics52\" align=\"BOTTOM\" width=\"8\" height=\"10\" border=\"0\"> is computed by taking the time derivative of the velocity:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image140.gif\" name=\"graphics53\" align=\"BOTTOM\" width=\"88\" height=\"34\" border=\"0\"><\/p>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image143.gif\" name=\"graphics54\" align=\"ABSMIDDLE\" width=\"80\" height=\"14\" border=\"0\">, we can substitute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image146.gif\" name=\"graphics55\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image149.gif\" name=\"graphics56\" align=\"BOTTOM\" width=\"38\" height=\"11\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image152.gif\" name=\"graphics57\" align=\"BOTTOM\" width=\"30\" height=\"11\" border=\"0\"> into the example formula to find <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p4_html_2d891da5.gif\" name=\"graphics58\" align=\"BOTTOM\" width=\"97\" height=\"25\" border=\"0\">. If you are familiar with physics, you may recognize <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p4_clip_image157.gif\" name=\"graphics59\" align=\"ABSMIDDLE\" width=\"8\" height=\"10\" border=\"0\"> as an approximation for the acceleration due to gravity.<\/p>\n<h3>Differentiability and Continuity<\/h3>\n<p>A function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> is continuous at the value <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image006.gif\" name=\"graphics4\" align=\"BOTTOM\" width=\"32\" height=\"7\" border=\"0\"> if and only if the following three rules hold.<\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image009.gif\" name=\"graphics5\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> exists.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image012.gif\" name=\"graphics6\" align=\"ABSMIDDLE\" width=\"53\" height=\"20\" border=\"0\"> exists.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image015.gif\" name=\"graphics7\" align=\"ABSMIDDLE\" width=\"102\" height=\"20\" border=\"0\">\n<\/li>\n<\/ul>\n<p>A function is discontinuous along an interval if there are any breaks in the graph. If the value of the function shifts at a point, if the function is not defined at a point, or if the function increases or decreases without bound at a point (as indicated by a vertical asymptote) the function is discontinuous.<\/p>\n<p>The slope (i.e. the derivative) of a curve is not always defined at all points on a continuous curve. A classic example of this is the absolute value function. Below is a graph of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p5_html_1ec3a16f.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"64\" height=\"27\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.2%20Art%20001.JPG\" name=\"graphics9\" alt=\" Graph of  y= the absolute value of x -1\" align=\"BOTTOM\" width=\"266\" height=\"250\" border=\"0\"><\/p>\n<p>What is the value of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image024.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">? It is clear that the function is defined at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image030.gif\" name=\"graphics12\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">. The derivative is a limit, and this limit must be defined in an open interval around the point <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image033.gif\" name=\"graphics13\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">. However, no matter how small we make the open interval around <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image036.gif\" name=\"graphics14\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">, the slope of the curve in the portion of the open interval to the left of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image039.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> will always be a negative number, and the slope of the curve in the portion of the open interval to the right of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image042.gif\" name=\"graphics16\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> will always be a positive number, so the limit is not defined since the limit from the right hand and left hand sides do not agree. You can also see this graphically. The tangent line must intersect the curve only at one point: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image045.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">. Prove to yourself that there are an infinite number of lines, all with different slopes, which could be described as a tangent line (any line that touches <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image048.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\"> but does not penetrate the \u2018V\u2019). Be wary whenever you see a sharp turn or \u201cpointy\u201d graph when evaluating derivatives because the derivative will not exist at these sharp points or \u201ccusps\u201d.<\/p>\n<p>Another example of an undefined derivative is at a point of \tdiscontinuity. If a function is discontinuous at a point, there is no derivative at that point.<\/p>\n<p>Some functions are not differentiable at a certain point because the derivative increases or decreases without bound. One example is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image051.gif\" name=\"graphics19\" align=\"ABSBOTTOM\" width=\"98\" height=\"28\" border=\"0\">. Below is the graph of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image054.gif\" name=\"graphics20\" align=\"ABSBOTTOM\" width=\"29\" height=\"14\" border=\"0\"> around <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image057.gif\" name=\"graphics21\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.2%20Art%20002.JPG\" name=\"graphics22\" alt=\"Graph of f(x) around x=0\" align=\"BOTTOM\" width=\"266\" height=\"250\" border=\"0\"><\/p>\n<p>At <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image060.gif\" name=\"graphics23\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, the slope becomes infinite or vertical, so <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image063.gif\" name=\"graphics24\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> is not defined there. In fact, we will later prove that any function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image066.gif\" name=\"graphics25\" align=\"ABSMIDDLE\" width=\"60\" height=\"15\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image069.gif\" name=\"graphics26\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">, will have an undefined derivative at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image072.gif\" name=\"graphics27\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">.<\/p>\n<p>We see that a function can be continuous at all points, while <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image075.gif\" name=\"graphics28\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> can be discontinuous at a point. This is , evidenced by \u201cpointy\u201d functions such as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p5_html_1ec3a16f.gif\" name=\"graphics29\" align=\"BOTTOM\" width=\"64\" height=\"27\" border=\"0\"> and functions whose derivatives have a vertical tangent line at a point such as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image081.gif\" name=\"graphics30\" align=\"ABSBOTTOM\" width=\"61\" height=\"28\" border=\"0\">. However, if a function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image084.gif\" name=\"graphics31\" align=\"ABSBOTTOM\" width=\"29\" height=\"14\" border=\"0\"> is differentiable at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image087.gif\" name=\"graphics32\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">, then it is also continuous at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p6_clip_image090.gif\" name=\"graphics33\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. In other words, differentiability always implies continuity, but continuity does not always imply differentiability.<\/p>\n<h3>How do we find derivatives of sums, products, and quotients?<\/h3>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>If two functions <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"28\" height=\"14\" border=\"0\"> are differentiable, the following rules apply.<\/p>\n<ul align=\"left\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p6_html_7fbc037.gif\" name=\"graphics5\" width=\"117\" height=\"27\" border=\"0\" align=\"absmiddle\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p6_html_m7b4e34a9.gif\" name=\"graphics6\" align=\"absmiddle\" width=\"123\" height=\"27\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image015.gif\" name=\"graphics7\" align=\"ABSMIDDLE\" width=\"173\" height=\"48\" border=\"0\"><\/li>\n<\/ul>\n<\/div>\n<p>These rules are proven below.<\/p>\n<p><strong>Proof for rule 1: <\/strong><\/p>\n<p align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image018.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"640\" height=\"101\" border=\"0\"><\/p>\n<p><strong>Proof for rule 2: <\/strong><\/p>\n<p align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"517\" height=\"189\" border=\"0\"><\/p>\n<p>We can prove rule 3, or the quotient rule, using the limit definition of derivatives as well. However, the quotient rule can be derived from the product rule (rule 2), by changing <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image024.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"12\" height=\"37\" border=\"0\"> to <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"26\" height=\"17\" border=\"0\">. Once we learn the general rule for differentiation of powers and an important rule for the derivative of composite functions known as the chain rule, the quotient rule can be easily derived from the product rule. Hence, it may not be necessary to memorize the quotient rule.<\/p>\n<p>Before we proceed with examples, let\u2019s learn three more important rules for derivatives:<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<ul align=\"left\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image030.gif\" name=\"graphics12\" align=\"ABSMIDDLE\" width=\"138\" height=\"14\" border=\"0\"> is a constant, then <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image033.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"87\" height=\"14\" border=\"0\">.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image036.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"60\" height=\"15\" border=\"0\">, then <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image039.gif\" name=\"graphics15\" align=\"ABSMIDDLE\" width=\"83\" height=\"17\" border=\"0\">, for <i>n<\/i> \u2260 0.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image042.gif\" name=\"graphics16\" align=\"ABSMIDDLE\" width=\"101\" height=\"35\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image045.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> is positive, then <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image048.gif\" name=\"graphics18\" align=\"ABSMIDDLE\" width=\"98\" height=\"17\" border=\"0\"> <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image051.gif\" name=\"graphics19\" align=\"BOTTOM\" width=\"42\" height=\"14\" border=\"0\">.<\/li>\n<\/ul>\n<\/div>\n<p><strong>Proof for rule 4: <\/strong><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image054.gif\" name=\"graphics20\" align=\"BOTTOM\" width=\"522\" height=\"77\" border=\"0\"><\/p>\n<p><strong>Proof for rule 5: <\/strong><br \/>\nWe can factor (<i>x<\/i> + \u0394<i>x<\/i>)<i><sup>n<\/sup><\/i> by using the Binomial Theorem.<\/p>\n<p class=\"lesson_text\">\n<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image057.gif\" name=\"graphics21\" align=\"BOTTOM\" width=\"326\" height=\"34\" border=\"0\">\n<\/p>\n<p>Look carefully at the numerator. You can see that the term in parentheses breaks down into an<i> x<\/i><i><sup>n<\/sup><\/i>-term (which cancels out) and a series of terms with <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image066.gif\" name=\"graphics24\" align=\"absmiddle\" width=\"111\" height=\"17\" border=\"0\"> etc. Since we divide the numerator by <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image069.gif\" name=\"graphics25\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\">, the only term we are interested in is the <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image072.gif\" name=\"graphics26\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\">-term. All the others will have <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image075.gif\" name=\"graphics27\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\"> to some power <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image078.gif\" name=\"graphics28\" align=\"BOTTOM\" width=\"18\" height=\"11\" border=\"0\">. Since the limit is for <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image081.gif\" name=\"graphics29\" align=\"BOTTOM\" width=\"46\" height=\"11\" border=\"0\">, all these will zero out. What we are left with is the coefficient in front of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image084.gif\" name=\"graphics30\" align=\"BOTTOM\" width=\"17\" height=\"11\" border=\"0\">. According to the Binomial Theorem, that term is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image087.gif\" name=\"graphics31\" align=\"BOTTOM\" width=\"31\" height=\"14\" border=\"0\">. Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image090.gif\" name=\"graphics32\" align=\"ABSMIDDLE\" width=\"83\" height=\"17\" border=\"0\">.<\/p>\n<p><strong>Proof for rule 6: <\/strong><br \/>\nApplying the quotient rule and the power rule (rule 5) we just derived, we find that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p7_clip_image093.gif\" name=\"graphics33\" align=\"BOTTOM\" width=\"354\" height=\"40\" border=\"0\"><\/p>\n<p>Let&#8217;s try an example. Given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"199\" height=\"35\" border=\"0\">, compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\">.<\/p>\n<p>We will apply several of the above rules to solve this derivative. First, rule 1 tells us that we can treat each of the 4 terms separately. We won&#8217;t need to separate terms like this in the future, but for now:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image009.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"129\" height=\"128\" border=\"0\"><\/p>\n<p>Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image012.gif\" name=\"graphics6\" align=\"BOTTOM\" width=\"247\" height=\"16\" border=\"0\">.<\/p>\n<p>The first term can be solved by combining rules 4 and 5.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"148\" height=\"19\" border=\"0\"><\/p>\n<p>The second term uses one of the memorized trigonometric derivatives.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image018.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"173\" height=\"16\" border=\"0\">.<\/p>\n<p>The third term combines rules 4 and 6.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"214\" height=\"35\" border=\"0\"><\/p>\n<p>The fourth term is easier to solve with the <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image024.gif\" name=\"graphics10\" align=\"BOTTOM\" width=\"19\" height=\"19\" border=\"0\">-term separated from the variable.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"289\" height=\"45\" border=\"0\">.<\/p>\n<p>Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image030.gif\" name=\"graphics12\" align=\"BOTTOM\" width=\"46\" height=\"14\" border=\"0\"> <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image033.gif\" name=\"graphics13\" align=\"BOTTOM\" width=\"29\" height=\"14\" border=\"0\"> <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image036.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"131\" height=\"39\" border=\"0\">.<\/p>\n<p>Let\u2019s try another example. Given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image039.gif\" name=\"graphics15\" align=\"ABSMIDDLE\" width=\"180\" height=\"17\" border=\"0\">, compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image042.gif\" name=\"graphics16\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\">.<\/p>\n<p>One method to compute <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image045.gif\" name=\"graphics17\" align=\"ABSMIDDLE\" width=\"33\" height=\"14\" border=\"0\"> is to factor the terms and derive a single polynomial expression. However, we can also use the product rule. Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image048.gif\" name=\"graphics18\" align=\"ABSMIDDLE\" width=\"122\" height=\"19\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image051.gif\" name=\"graphics19\" align=\"ABSMIDDLE\" width=\"86\" height=\"19\" border=\"0\">. Terefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image054.gif\" name=\"graphics20\" align=\"ABSMIDDLE\" width=\"92\" height=\"16\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image057.gif\" name=\"graphics21\" align=\"ABSMIDDLE\" width=\"74\" height=\"19\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image060.gif\" name=\"graphics22\" align=\"BOTTOM\" width=\"293\" height=\"89\" border=\"0\"><\/p>\n<p>Try one final example. Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image063.gif\" name=\"graphics23\" align=\"ABSMIDDLE\" width=\"204\" height=\"16\" border=\"0\">. If at least one of the terms can be expressed as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image066.gif\" name=\"graphics24\" align=\"ABSMIDDLE\" width=\"68\" height=\"17\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image069.gif\" name=\"graphics25\" align=\"BOTTOM\" width=\"6\" height=\"7\" border=\"0\"> is a constant and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image072.gif\" name=\"graphics26\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">, show that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image075.gif\" name=\"graphics27\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> is not differentiable at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image078.gif\" name=\"graphics28\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">.<\/p>\n<p>The derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image081.gif\" name=\"graphics29\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> is the sum of the derivatives of the terms, including <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image084.gif\" name=\"graphics30\" align=\"ABSMIDDLE\" width=\"30\" height=\"16\" border=\"0\">. From rules <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image087.gif\" name=\"graphics31\" align=\"BOTTOM\" width=\"7\" height=\"11\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image090.gif\" name=\"graphics32\" align=\"BOTTOM\" width=\"6\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image093.gif\" name=\"graphics33\" align=\"ABSMIDDLE\" width=\"139\" height=\"19\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image096.gif\" name=\"graphics34\" align=\"BOTTOM\" width=\"39\" height=\"11\" border=\"0\"> is a constant. Since <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image099.gif\" name=\"graphics35\" align=\"BOTTOM\" width=\"29\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image102.gif\" name=\"graphics36\" align=\"BOTTOM\" width=\"51\" height=\"11\" border=\"0\">. That means the exponent in the derivative expression is negative. Therefore, we can rewrite the derivative as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image105.gif\" name=\"graphics37\" align=\"ABSMIDDLE\" width=\"173\" height=\"35\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image108.gif\" name=\"graphics38\" align=\"ABSMIDDLE\" width=\"55\" height=\"14\" border=\"0\"> is a positive number. At <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image111.gif\" name=\"graphics39\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p8_clip_image114.gif\" name=\"graphics40\" align=\"ABSMIDDLE\" width=\"34\" height=\"16\" border=\"0\"> does not exist, since we have division by zero. Since at least one term in the function is not differentiable, the whole expression is not differentiable.<\/p>\n<h3>What is a composite function?<\/h3>\n<p>A <abbr rel=\"tooltip\" title=\"a function of a function; given two functions  MathType and  MathType, the composite function is denoted as  MathType or  MathType.\">composite function<\/abbr> is a function of a function. Given two functions <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"29\" height=\"14\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"28\" height=\"14\" border=\"0\">, the composite function is often denoted as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image009.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"64\" height=\"21\" border=\"0\"> or <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image012.gif\" name=\"graphics6\" align=\"BOTTOM\" width=\"50\" height=\"14\" border=\"0\">. For example, the function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"90\" height=\"21\" border=\"0\"> can be considered a composite function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image018.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"94\" height=\"14\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"67\" height=\"20\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image024.gif\" name=\"graphics10\" align=\"ABSMIDDLE\" width=\"79\" height=\"17\" border=\"0\">. Alternatively, we can have <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"87\" height=\"20\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image030.gif\" name=\"graphics12\" align=\"ABSMIDDLE\" width=\"58\" height=\"17\" border=\"0\">. Another composite function is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image033.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"124\" height=\"17\" border=\"0\">, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image036.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"74\" height=\"14\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image039.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"97\" height=\"17\" border=\"0\">.<\/p>\n<p>According to the <abbr rel=\"tooltip\" title=\"a theorem for computing the derivative of a composite function (see composite function). If MathType are differentiable functions, and for every point p such that  MathType is defined,  MathType is also defined, then the derivative of a composite function MathType is  MathType.\">chain rule<\/abbr>, the derivative of a composite function <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image042.gif\" name=\"graphics16\" align=\"ABSMIDDLE\" width=\"94\" height=\"14\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image045.gif\" name=\"graphics17\" align=\"ABSMIDDLE\" width=\"145\" height=\"14\" border=\"0\">.<\/p>\n<p>A couple of conditions apply to this rule.<\/p>\n<ul>\n<li>For every point <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image048.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"9\" height=\"10\" border=\"0\">, such that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image051.gif\" name=\"graphics19\" align=\"BOTTOM\" width=\"29\" height=\"14\" border=\"0\"> is defined, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image054.gif\" name=\"graphics20\" align=\"ABSMIDDLE\" width=\"52\" height=\"14\" border=\"0\"> is also defined.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image057.gif\" name=\"graphics21\" align=\"ABSMIDDLE\" width=\"50\" height=\"14\" border=\"0\"> are differentiable functions.<\/li>\n<\/ul>\n<p>Here is another way to write the chain rule.<\/p>\n<p align=\"CENTER\">\nLet <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image060.gif\" name=\"graphics22\" align=\"ABSMIDDLE\" width=\"53\" height=\"14\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image063.gif\" name=\"graphics23\" align=\"ABSMIDDLE\" width=\"77\" height=\"14\" border=\"0\">.\n<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image066.gif\" name=\"graphics24\" align=\"ABSMIDDLE\" width=\"141\" height=\"34\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image069.gif\" name=\"graphics25\" align=\"ABSMIDDLE\" width=\"68\" height=\"34\" border=\"0\">.<\/p>\n<p>Therefore, we write the chain rule as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image072.gif\" name=\"graphics26\" align=\"ABSMIDDLE\" width=\"73\" height=\"34\" border=\"0\">.<\/p>\n<p>Before we prove the chain rule, let\u2019s do an example. Given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image075.gif\" name=\"graphics27\" align=\"ABSMIDDLE\" width=\"122\" height=\"17\" border=\"0\">, find <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image078.gif\" name=\"graphics28\" align=\"ABSMIDDLE\" width=\"17\" height=\"34\" border=\"0\">. Recalling the rule for the derivative of powers, you might be tempted to say <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image081.gif\" name=\"graphics29\" align=\"ABSMIDDLE\" width=\"143\" height=\"34\" border=\"0\">, but this would be incomplete. According to the chain rule, we must now multiply this term with the derivative of the term inside the parentheses. To clarify this, let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image084.gif\" name=\"graphics30\" align=\"BOTTOM\" width=\"100\" height=\"14\" border=\"0\">. Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image087.gif\" name=\"graphics31\" align=\"ABSMIDDLE\" width=\"43\" height=\"17\" border=\"0\">, so <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image090.gif\" name=\"graphics32\" align=\"ABSMIDDLE\" width=\"87\" height=\"34\" border=\"0\">, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image093.gif\" name=\"graphics33\" align=\"ABSMIDDLE\" width=\"65\" height=\"34\" border=\"0\">. According to the chain rule,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image096.gif\" name=\"graphics34\" align=\"BOTTOM\" width=\"377\" height=\"34\" border=\"0\">.<\/p>\n<p><span style=\"text-decoration: none\">After enough examples applying the chain rule, it should become second nature. It is similar to taking the derivative of the <\/span><em><span style=\"text-decoration: none\">outside<\/span><\/em><span style=\"text-decoration: none\"> function and multiplying it by the derivative of the <\/span><em><span style=\"text-decoration: none\">inside<\/span><\/em><span style=\"text-decoration: none\"> part. The chain rule will be used to calculate derivatives of trigonometric, logarithmic, exponential, and other function types, once we learn how to compute their derivatives. <\/span><\/p>\n<h3>Proof for the Chain Rule:<\/h3>\n<p>For this proof, we will assume <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image099.gif\" name=\"graphics35\" align=\"ABSMIDDLE\" width=\"165\" height=\"14\" border=\"0\">. This assumption is not necessary for the chain rule, but it makes it simpler to prove. Many calculus textbooks extend the proof for the case <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image102.gif\" name=\"graphics36\" align=\"BOTTOM\" width=\"41\" height=\"11\" border=\"0\">.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image105.gif\" name=\"graphics37\" align=\"BOTTOM\" width=\"638\" height=\"90\" border=\"0\"><\/p>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image108.gif\" name=\"graphics38\" align=\"ABSMIDDLE\" width=\"8\" height=\"10\" border=\"0\"> is differentiable, we know it is also continuous. Therefore, as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image111.gif\" name=\"graphics39\" align=\"BOTTOM\" width=\"46\" height=\"11\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image114.gif\" name=\"graphics40\" align=\"ABSMIDDLE\" width=\"113\" height=\"14\" border=\"0\">. Now we define <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image117.gif\" name=\"graphics41\" align=\"ABSMIDDLE\" width=\"310\" height=\"14\" border=\"0\">. Note that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image120.gif\" name=\"graphics42\" align=\"BOTTOM\" width=\"490\" height=\"25\" border=\"0\">.<\/p>\n<p>Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image123.gif\" name=\"graphics43\" align=\"ABSMIDDLE\" width=\"117\" height=\"14\" border=\"0\">, so we can continue our proof.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p9_clip_image126.gif\" name=\"graphics44\" align=\"BOTTOM\" width=\"540\" height=\"62\" border=\"0\"><\/p>\n<p>Let\u2019s try a few more examples applying the chain rule. See if you can solve for the derivative without breaking the composite function into two functions.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>In finding the derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image003.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" width=\"130\" height=\"17\" border=\"0\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image006.gif\" name=\"graphics4\" align=\"ABSMIDDLE\" width=\"46\" height=\"14\" border=\"0\"><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image009.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"142\" height=\"17\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image012.gif\" name=\"graphics6\" align=\"BOTTOM\" width=\"154\" height=\"17\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"85\" height=\"17\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image018.gif\" name=\"graphics8\" align=\"BOTTOM\" width=\"97\" height=\"17\" border=\"0\"><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\" style=\"display: none;\">\n<p>The correct choice is A. First, recall that the derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image021.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"28\" height=\"11\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image024.gif\" name=\"graphics10\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">. Then apply the chain rule.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image027.gif\" name=\"graphics11\" align=\"BOTTOM\" width=\"382\" height=\"17\" border=\"0\">.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>In finding the derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image031.gif\" name=\"graphics12\" width=\"112\" height=\"36\" border=\"0\" align=\"ABSMIDDLE\">, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image033.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"46\" height=\"14\" border=\"0\"><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image037.gif\" name=\"graphics14\" align=\"BOTTOM\" width=\"74\" height=\"36\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image039.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"39\" height=\"34\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image043.gif\" name=\"graphics16\" align=\"BOTTOM\" width=\"102\" height=\"36\" border=\"0\"><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image044.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"93\" height=\"72\" border=\"0\"><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\" style=\"display: none;\">\n<p><span class=\"lesson_text\">The correct choice is D. Apply the chain rule to solve the derivative. (Remember that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image048.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"50\" height=\"26\" border=\"0\">). <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image049.gif\" name=\"graphics19\" width=\"282\" height=\"72\" border=\"0\" align=\"BOTTOM\">.<\/p>\n<\/div>\n<\/section>\n<p>Here is another more complicated example below:<\/p>\n<p>Given <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image054.gif\" name=\"graphics20\" align=\"ABSMIDDLE\" width=\"108\" height=\"48\" border=\"0\">, find <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image057.gif\" name=\"graphics21\" align=\"ABSMIDDLE\" width=\"27\" height=\"14\" border=\"0\">. You may want to break up the composite function in this example, because it is more complicated. Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image060.gif\" name=\"graphics22\" align=\"ABSMIDDLE\" width=\"89\" height=\"40\" border=\"0\">, so <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image063.gif\" name=\"graphics23\" align=\"ABSMIDDLE\" width=\"53\" height=\"17\" border=\"0\">, and solve <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image066.gif\" name=\"graphics24\" align=\"ABSMIDDLE\" width=\"80\" height=\"34\" border=\"0\">.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image069.gif\" name=\"graphics25\" align=\"ABSMIDDLE\" width=\"149\" height=\"48\" border=\"0\"> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image072.gif\" name=\"graphics26\" align=\"ABSMIDDLE\" width=\"17\" height=\"34\" border=\"0\"> can be solved using the quotient rule.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image075.gif\" name=\"graphics27\" align=\"BOTTOM\" width=\"365\" height=\"40\" border=\"0\"><\/p>\n<p>Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p10_clip_image078.gif\" name=\"graphics28\" align=\"ABSMIDDLE\" width=\"584\" height=\"48\" border=\"0\"><\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The <strong><em>limit<\/em><\/strong> of a function is the value the function approaches as its variable approaches a number to the left and right of that \tnumber. The function need not be defined at that number, but must be defined for values slightly less and slightly more. Limits can be determined (if they exist) by plugging in the exact value and values on either side of the value itself, by graphing the function, or by mathematical manipulation such as factoring.<\/li>\n<li>A <em><strong>one-sided limit<\/strong><\/em> approaches a value to one side of the variable but not the other. If a function is defined for numbers slightly higher than <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image003.gif\" name=\"graphics3\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> but not for numbers slightly lower, the function has a right-hand limit, and we say that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image006.gif\" name=\"graphics4\" align=\"MIDDLE\" width=\"57\" height=\"23\" border=\"0\"> exists.<\/li>\n<li><em><strong>Infinite limits<\/strong><\/em> are limits that approach <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p10_html_65cbb539.gif\" name=\"graphics5\" align=\"BOTTOM\" width=\"25\" height=\"16\" border=\"0\"> as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image012.gif\" name=\"graphics6\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> approaches <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image015.gif\" name=\"graphics7\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\">. This is represented graphically with a vertical asymptote at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image018.gif\" name=\"graphics8\" align=\"ABSMIDDLE\" width=\"32\" height=\"7\" border=\"0\">.<\/li>\n<li>A <strong><em>limit at infinity<\/em><\/strong> is the value of a function as <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image020.gif\" name=\"graphics9\" align=\"BOTTOM\" width=\"7\" height=\"7\" border=\"0\"> approaches <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s2_p10_html_65cbb539.gif\" name=\"graphics10\" align=\"BOTTOM\" width=\"25\" height=\"16\" border=\"0\">. For example, if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image025.gif\" name=\"graphics11\" align=\"ABSMIDDLE\" width=\"80\" height=\"20\" border=\"0\">, the limit at infinity is represented with a horizontal asymptote at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image028.gif\" name=\"graphics12\" align=\"ABSMIDDLE\" width=\"35\" height=\"14\" border=\"0\">.<\/li>\n<li>A function is <em><strong>continuous<\/strong><\/em> over an interval if there are no breaks in the graph. In other words, for all points <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image030.gif\" name=\"graphics13\" align=\"ABSMIDDLE\" width=\"7\" height=\"7\" border=\"0\"> in the interval (except the endpoints if the interval is open), <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image033.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" width=\"102\" height=\"20\" border=\"0\">, and terms to both the left and right of the equal sign exist.<\/li>\n<li>The <strong><em>Intermediate Value Theorem<\/em><\/strong> states that, if a function is continuous over a closed interval, every value of the function between the endpoints exists at least once, and there is a value of the variable within the open range that defines the function\u2019s value.<\/li>\n<li>The derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image036.gif\" name=\"graphics15\" align=\"BOTTOM\" width=\"28\" height=\"11\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image039.gif\" name=\"graphics16\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\">.<\/li>\n<li>The derivative of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image041.gif\" name=\"graphics17\" align=\"BOTTOM\" width=\"31\" height=\"11\" border=\"0\"> is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p5_clip_image044.gif\" name=\"graphics18\" align=\"BOTTOM\" width=\"40\" height=\"11\" border=\"0\">.<\/li>\n<li>A line that intersects a curve at two points is known as a <em><strong>secant line.<\/strong><\/em><\/li>\n<li>A line that intersects a curve at just one point is known as a <em><strong>tangent line<\/strong><\/em>.<\/li>\n<\/ul>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/limits-and-continuity\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus-ii\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next\u00a0Lesson \u27a1 Differential Calculus Objective In this lesson, you will cover the definition of a derivative and apply it to the evaluation of slopes and tangent lines of curves, instantaneous rates of changes of functions, relative maxima and minima, and methods to compute the derivative of several types of functions. Previously Covered: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-333","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/333","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=333"}],"version-history":[{"count":57,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/333\/revisions"}],"predecessor-version":[{"id":994,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/333\/revisions\/994"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}