// Numbas version: exam_results_page_options
{"name": "Helge's copy of DIfferentiation: product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"type": "question", "rulesets": {"dpoly": ["std", "fractionNumbers"], "std": ["all", "!collectNumbers"]}, "parts": [{"type": "gapfill", "gaps": [{"checkingtype": "absdiff", "type": "jme", "answersimplification": "dPoly", "expectedvariablenames": [], "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "vsetrange": [0, 1], "showpreview": true, "showCorrectAnswer": true, "vsetrangepoints": 5, "checkvariablenames": false, "scripts": {}, "checkingaccuracy": 0.001, "marks": 3}], "showCorrectAnswer": true, "steps": [{"checkingtype": "absdiff", "type": "jme", "expectedvariablenames": [], "answer": "", "vsetrange": [0, 1], "showpreview": true, "showCorrectAnswer": true, "vsetrangepoints": 5, "checkvariablenames": false, "scripts": {}, "checkingaccuracy": 0.001, "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}], "scripts": {}, "stepsPenalty": 0, "prompt": "\n        <p>$\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n        <p>You are given that \\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]</p>\n        <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Click on Show steps to get more information, you will not lose any marks by doing so.</p>\n        ", "marks": 0}], "functions": {}, "variable_groups": [], "variables": {"m": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8)", "name": "m"}, "a": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "a"}, "b": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..5)", "name": "b"}, "n": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)", "name": "n"}, "s1": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1"}}, "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Helge's copy of DIfferentiation: product rule", "ungrouped_variables": ["a", "s1", "b", "m", "n"], "showQuestionGroupNames": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "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 $f^{\\prime}(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International", "notes": "\n                        \t\t<p><strong>20/06/2012:</strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p><strong>4/7/2012<br /></strong></p>\n                        \t\t<p>Added tags.</p>\n                        \t\t<p><strong>31/07/2012:</strong></p>\n                        \t\t<p>Added a tag.</p>\n                        \t\t<p>Allowed free look at Steps.</p>\n                        \t\t"}, "advice": "\n    \n    \n    <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    \n    \n    \n    <p>For this example:</p>\n    \n    \n    \n    <p>\\[\\simplify[dPoly]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[dPoly]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \n    <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    \n    \n    \n    <p>The last step was to take out the common term $\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.</p>\n    \n    \n    \n    <p>Hence \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].</p>\n    \n    \n    ", "tags": ["algebraic manipulation", "Calculus", "calculus", "checked2015", "derivative", "derivative ", "deriving a function", "differentiate", "differentiating a function", "differentiating a product of functions", "differentiation", "exponential function", "functions", "mas1601", "MAS1601", "product rule", "Steps", "steps"], "contributors": [{"name": "Helge M\u00fcnnich", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1885/"}]}]}], "contributors": [{"name": "Helge M\u00fcnnich", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1885/"}]}