// 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": [{"name": "Product rule", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Product rule</p>"}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "<p>Differentiate the following using the product rule:</p>\n<p style=\"padding-left: 30px;\">\\(y=\\var{g}x^{\\var{a}} \\times (\\var{c}x^{\\var{d}} + \\var{b})\\)</p>", "variable_groups": [], "functions": {}, "variables": {"g": {"name": "g", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(4 .. 15#1)"}, "b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2 .. 5#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2 .. 12#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2 .. 9#0.25)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(3 .. 9#1)"}}, "tags": [], "parts": [{"variableReplacements": [], "gaps": [{"checkingAccuracy": 0.001, "variableReplacements": [], "vsetRangePoints": 5, "vsetRange": [0, 1], "valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "useCustomName": false, "failureRate": 1, "checkingType": "absdiff", "checkVariableNames": false, "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showPreview": true, "answer": "{a}{g}x^{a-1}({c}x^{d}+{b})  +  {g}x^{a} ({c}{d}x^{d-1}+{b})", "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": "3"}], "type": "gapfill", "useCustomName": false, "prompt": "<p>\\(\\frac{dy}{dx}=\\)[[0]]</p>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>\\(y= \\var{g}x^{\\var{a}} \\times (\\var{c}x^{\\var{d}} + \\var{b})\\)</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 = \\var{g}x^{\\var{a}}\\)&nbsp; and &nbsp;\\(v=\\var{c}x^{\\var{d}} + \\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{du}{dx}=\\var{a}\\var{g}x^\\var{a-1}\\) &nbsp;and &nbsp;\\(\\frac{dv}{dx}=\\var{c}\\var{d}x^{\\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{d}} + \\var{b}) \\times \\var{a}\\var{g}x^\\var{a-1} +&nbsp; \\var{g}x^\\var{a} \\times (\\var{c}\\var{d}x^{\\var{d-1}} + \\var{b})\\)</p>\n<p style=\"padding-left: 30px;\">Which tidies up to&nbsp;\\(\\frac{dy}{dx}= \\var{a}\\var{g}x^\\var{a-1}(\\var{c}x^{\\var{d}} + \\var{b})&nbsp; +&nbsp; \\var{g}x^\\var{a} (\\var{c}\\var{d}x^{\\var{d-1}} + \\var{b})\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "rulesets": {}, "ungrouped_variables": ["a", "b", "c", "d", "g"], "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/"}]}