// Numbas version: exam_results_page_options
{"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "statement": "<p>Differentiate the following using the product rule:</p>\n<p style=\"padding-left: 30px;\">\\(y=x^{\\var{a}} \\times (\\var{c}x + \\var{b})^{\\var{d}}\\)</p>", "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d", "a5"], "variables": {"a5": {"group": "Ungrouped variables", "description": "", "definition": "random(4 .. 15#1)", "name": "a5", "templateType": "randrange"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "random(2 .. 9#1)", "name": "a", "templateType": "randrange"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "random(2 .. 5#1)", "name": "b", "templateType": "randrange"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "random(2 .. 12#1)", "name": "c", "templateType": "randrange"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "random(3 .. 9#1)", "name": "d", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Product rule</p>"}, "parts": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "customMarkingAlgorithm": "", "customName": "", "prompt": "<p>\\(\\frac{dy}{dx}=\\)[[0]]</p>", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"failureRate": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": "3", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "customName": "", "answer": "{a}x^{a-1}({c}x+{b})^{d} + {c}{d}x^{a}({c}x + {b})^{d-1}", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "checkVariableNames": false, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "type": "jme", "vsetRange": [0, 1], "useCustomName": false, "showFeedbackIcon": true, "showPreview": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacements": [], "scripts": {}}], "extendBaseMarkingAlgorithm": true, "type": "gapfill", "useCustomName": false, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}}], "advice": "<p>\\(y=x^{\\var{a}} \\times (\\var{c}x + \\var{b})^{\\var{d}}\\)</p>\n<p>Recall the product rule if \\(y=u \\times v\\) where \\(u\\) and \\(v\\) are both functions of \\(x\\) then</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dy}{dx}=v \\times \\frac{du}{dx}+u \\times \\frac{dv}{dx}\\)</p>\n<p style=\"padding-left: 30px;\">let \\( u = x^{\\var{a}}\\)&nbsp; and &nbsp;\\(v=(\\var{c}x + \\var{b})^{\\var{d}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{du}{dx}=\\var{a}x^\\var{a-1}\\) &nbsp;and &nbsp;\\(\\frac{dv}{dx}=\\var{c}\\var{d}(\\var{c}x + \\var{b})^{\\var{d-1}}\\)</p>\n<p style=\"padding-left: 30px;\">Substituting into&nbsp;\\(\\frac{dy}{dx}=v \\times \\frac{du}{dx}+u \\times \\frac{dv}{dx}\\) gives</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dy}{dx}=(\\var{c}x + \\var{b})^{\\var{d}} \\times \\var{a}x^\\var{a-1} +&nbsp; x^\\var{a} \\times \\var{c}\\var{d}(\\var{c}x + \\var{b})^{\\var{d-1}}\\)</p>\n<p style=\"padding-left: 30px;\">Which tidies up to&nbsp;\\(\\frac{dy}{dx}=\\var{a}x^\\var{a-1}(\\var{c}x + \\var{b})^{\\var{d}}&nbsp; + \\var{c}\\var{d}x^\\var{a}(\\var{c}x + \\var{b})^{\\var{d-1}}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "functions": {}, "rulesets": {}, "extensions": [], "name": "Product rule", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}