// Numbas version: exam_results_page_options
{"name": "JD'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": [{"parts": [{"extendBaseMarkingAlgorithm": true, "gaps": [{"checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "showFeedbackIcon": true, "answerSimplification": "std", "scripts": {}, "failureRate": 1, "vsetRange": [0, 1], "valuegenerators": [{"name": "x", "value": ""}], "checkingAccuracy": 0.001, "useCustomName": false, "showPreview": true, "extendBaseMarkingAlgorithm": true, "type": "jme", "customName": "", "vsetRangePoints": 5, "customMarkingAlgorithm": "", "unitTests": [], "marks": 3, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacements": []}], "steps": [{"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>", "type": "information", "variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "marks": 0, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": []}], "stepsPenalty": 0, "prompt": "\n      <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>\n      ", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "useCustomName": false, "variableReplacements": []}], "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $f(x) = x^m(a x+b)^n$.</p>"}, "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  ", "rulesets": {"surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "preamble": {"js": "", "css": ""}, "functions": {}, "tags": [], "extensions": [], "variables": {"b": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "b", "definition": "s1*random(1..9)"}, "s1": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "s1", "definition": "random(1,-1)"}, "n": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "n", "definition": "random(3..9)"}, "m": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "m", "definition": "random(3..9)"}, "a": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "a", "definition": "random(2..9)"}}, "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "JD's copy of Differentiation: Product rule", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "JD Ichwan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3389/"}]}