// Numbas version: exam_results_page_options
{"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Quotient rule</p>"}, "type": "question", "preamble": {"css": "", "js": ""}, "advice": "<p>\\(f(x)=\\frac{\\var{a} +\\var{g}x^{\\var{b}}}{\\var{c}x^{\\var{d}} +\\var{f}}\\)</p>\n<p>Recall the quotient rule: if \\(y=\\frac{u}{v}\\) where \\(u\\) and \\(v\\) are both functions of \\(x\\) then</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}-u\\frac{dv}{dx}}{v^2}\\)</p>\n<p>Let &nbsp; \\(u=\\var{a} +\\var{g}x^{\\var{b}}\\) &nbsp; and &nbsp; \\(v=\\var{c}x^{\\var{d}} +\\var{f}\\)</p>\n<p>then &nbsp; \\(\\frac{du}{dx}=\\var{g}\\var{b}x^{\\var{b-1}}\\) &nbsp; and &nbsp; \\(\\frac{dv}{dx}=\\var{d}\\var{c}x^{\\var{d-1}}\\)</p>\n<p>Putting these results together as shown in the rule gives:</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\frac{(\\var{c}x^{\\var{d}} +\\var{f}) \\times \\var{g}\\var{b}x^{\\var{b-1}} - (\\var{a} +\\var{g}x^{\\var{b}})&nbsp;\\times \\var{d}\\var{c}x^{\\var{d-1}}}{(\\var{c}x^{\\var{d}} +\\var{f})^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\frac{\\var{g}\\var{b}x^{\\var{b-1}}(\\var{c}x^{\\var{d}} +\\var{f}) - \\var{d}\\var{c}x^{\\var{d-1}}(\\var{a} +\\var{g}x^{\\var{b}}) }{(\\var{c}x^{\\var{d}} +\\var{f})^2}\\)</p>", "rulesets": {}, "extensions": [], "name": "Quotient rule", "ungrouped_variables": ["a", "b", "c", "f", "d", "g"], "functions": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"b": {"name": "b", "definition": "random(2..6#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "a": {"name": "a", "definition": "random(2..10#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "d": {"name": "d", "definition": "random(2..7#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "g": {"name": "g", "definition": "random(-10..-2#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "f": {"name": "f", "definition": "random(1..11#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "c": {"name": "c", "definition": "random(2..12#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}}, "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\frac{\\var{a} + \\var{g}x^{\\var{b}}}{\\var{c}x^{\\var{d}} + \\var{f}}\\)</p>", "parts": [{"type": "gapfill", "customMarkingAlgorithm": "", "unitTests": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"type": "jme", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "showPreview": true, "answer": "({b}{g}x^{b-1}({c}x^{d} + {f})-{d}{c}x^{d-1}({a} + {g}x^{b}))/({c}x^{d} +{f})^2", "marks": "3", "unitTests": [], "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "variableReplacements": [], "checkingAccuracy": 0.001, "checkingType": "absdiff", "showCorrectAnswer": true, "showFeedbackIcon": true, "failureRate": 1, "vsetRange": [0, 1]}], "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "prompt": "<p>\\(\\frac{df}{dx} = \\)&nbsp;[[0]]</p>", "variableReplacements": []}], "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/"}]}