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{"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "f", "d"], "name": "Quotient rule", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>\\(f(x)=\\frac{\\var{a}x^{\\var{b}}+\\var{f}}{\\var{c}cos(\\var{d}x)}\\)</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}x^{\\var{b}}+\\var{f}\\) &nbsp; and &nbsp; \\(v=\\var{c}cos(\\var{d}x)\\)</p>\n<p>then &nbsp; \\(\\frac{du}{dx}=\\var{b}*\\var{a}x^{\\var{b}-1}\\) &nbsp; and &nbsp; \\(\\frac{dv}{dx}=-\\var{d}*\\var{c}sin(\\var{d}x)\\)</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}cos(\\var{d}x))*\\var{b}*\\var{a}x^{\\var{b}-1}-(\\var{a}x^{\\var{b}}+\\var{f})*(-\\var{d}*\\var{c}sin(\\var{d}x))}{(\\var{c}cos(\\var{d}x))^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\frac{\\simplify{({c}*cos({d}x))*{b}*{a}x^{{b}-1}+({a}x^{{b}}+{f})*({c}*{d}*sin({d}x))}}{(\\var{c}*cos(\\var{d}x))^2}\\)</p>", "rulesets": {}, "parts": [{"prompt": "<p>\\(\\frac{df}{dx} = \\)&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}*cos({d}*x)*{a}*{b}*x^({b}-1)+({a}*x^{b}+{f})*{c}*{d}*sin({d}*x))/({c}*cos({d}*x))^2", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\frac{\\var{a}x^{\\var{b}}+\\var{f}}{\\var{c}cos(\\var{d}x)}\\)</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..6#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(2..7#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d", "description": ""}, "f": {"definition": "random(1..11#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "f", "description": ""}}, "metadata": {"description": "<p>Quotient rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}