// Numbas version: exam_results_page_options
{"name": "Harry's copy of Differentation: Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showFeedbackIcon": true, "steps": [{"showFeedbackIcon": true, "variableReplacements": [], "prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "scripts": {}}], "variableReplacements": [], "prompt": "\n\t\t\t<p>$\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n\t\t\t      <p>You are given that \\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]</p>\n\t\t\t      <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n\t\t\t      <p>$g(x)=\\;$[[0]]</p>\n\t\t\t      <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n\t\t\t", "stepsPenalty": 0, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "variableReplacements": [], "checkingtype": "absdiff", "marks": 3, "variableReplacementStrategy": "originalfirst", "type": "jme", "vsetrange": [0, 1], "showpreview": true, "showFeedbackIcon": true, "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "answersimplification": "dPoly", "expectedvariablenames": [], "scripts": {}, "showCorrectAnswer": true, "vsetrangepoints": 5, "checkingaccuracy": 0.001}], "scripts": {}}], "rulesets": {"dPoly": ["fractionNumbers", "std"]}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "statement": "<p><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">The product rule states that if $u$ and $v$ are functions of $x$ then</span><br style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"/><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</span></p>\n<p>Differentiate the following function $f(x)$ using the product rule.</p>", "extensions": [], "tags": [], "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variables": {"s1": {"description": "", "name": "s1", "definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "definition": "s1*random(1..5)", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables"}, "n": {"description": "", "name": "n", "definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables"}, "m": {"description": "", "name": "m", "definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables"}}, "name": "Harry's copy of Differentation: Product Rule", "advice": "\n\t  \n\t  \n\t  <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>For this example:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[dPoly]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[dPoly]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>Hence on substituting into the product rule above we get:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>The last step was to take out the common term $\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.</p>\n\t  \n\t  \n\t  \n\t  <p>Hence \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].</p>\n\t  \n\t  \n\t  \n\t", "metadata": {"description": "<p>Differentiate the function $f(x)=(a + b x)^m &nbsp;e ^ {n x}$&nbsp;using the product rule. Find $g(x)$ such that $f\\;'(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}