// Numbas version: exam_results_page_options
{"name": "Heather'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": [{"statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variables": {"s1": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s1", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "random(2..9)", "name": "a", "templateType": "anything"}, "n": {"group": "Ungrouped variables", "description": "", "definition": "random(3..9)", "name": "n", "templateType": "anything"}, "m": {"group": "Ungrouped variables", "description": "", "definition": "random(3..9)", "name": "m", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "s1*random(1..9)", "name": "b", "templateType": "anything"}}, "extensions": [], "variable_groups": [], "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "parts": [{"marks": 0, "customMarkingAlgorithm": "", "stepsPenalty": 0, "prompt": "<p>$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$</p>\n<p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n<p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>", "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"marks": "2", "customMarkingAlgorithm": "", "expectedVariableNames": [], "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "checkingType": "absdiff", "showFeedbackIcon": true, "showCorrectAnswer": true, "failureRate": 1, "checkingAccuracy": 0.001, "showPreview": true, "extendBaseMarkingAlgorithm": true, "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "type": "jme", "checkVariableNames": false, "answerSimplification": "std", "scripts": {}, "variableReplacements": []}], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "scripts": {}, "variableReplacements": [], "steps": [{"marks": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "variableReplacementStrategy": "originalfirst", "type": "information", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "variableReplacements": []}]}], "preamble": {"js": "", "css": ""}, "advice": "\n  \n  \n  <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{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[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  ", "ungrouped_variables": ["a", "s1", "b", "m", "n"], "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "name": "Heather's copy of Differentiation : Product Rule", "metadata": {"description": "<p>Differentiate $f(x) = x^m(a x+b)^n$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}, {"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}, {"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}]}