// Numbas version: finer_feedback_settings
{"name": "Quotient rule - differentiate quadratic over quadratic", "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>The derivative of $\\displaystyle \\frac{ax^2+b}{cx^2+d}$ is $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.</p>"}, "preamble": {"css": "", "js": ""}, "advice": "\n  \n  \n  <p>The quotient rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]</p>\n  \n  \n  \n  <p>For this example:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{u = ({a}x^2+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {2*a}x}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({c} * x^2+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the quotient rule above we get:</p>\n  \n  \n  \n  <p>\\[\\begin{eqnarray*} \\frac{df}{dx}&#38;=&#38;\\simplify[std]{({2*a}x({c}x^2+{d})-{2*c}x({a}x^2+{b}))/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({2*a*c}x^3+{2*a*d}x-{2*c*a}x^3-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({2*det}x)/({c}x^2+{d})^2}\n  \n  \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{2*det}x}$</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "name": "Quotient rule - differentiate quadratic over quadratic", "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "functions": {}, "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "det": {"name": "det", "group": "Ungrouped variables", "definition": "a*d-b*c", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "templateType": "anything"}}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "parts": [{"customName": "", "type": "gapfill", "customMarkingAlgorithm": "", "steps": [{"customName": "", "type": "information", "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "prompt": "<p>The quotient rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]</p>", "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}], "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"customName": "", "type": "jme", "vsetRangePoints": 5, "answer": "{2*det}x", "showFeedbackIcon": true, "failureRate": 1, "checkingType": "absdiff", "checkVariableNames": false, "notallowed": {"showStrings": false, "message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "partialCredit": 0, "strings": ["."]}, "showCorrectAnswer": true, "useCustomName": false, "customMarkingAlgorithm": "", "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "marks": 3, "valuegenerators": [{"name": "x", "value": ""}], "checkingAccuracy": 0.001, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showPreview": true, "unitTests": [], "variableReplacements": [], "vsetRange": [0, 1]}], "sortAnswers": false, "prompt": "\n      <p>\\[\\simplify[std]{f(x) = ({a} * x^2+{b})/({c}x^2+{d})}\\]<br/>You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d})^2}\\]<br/>for a polynomial $g(x)$.&nbsp;You are asked to find $g(x)$</p>\n      <p>$g(x)=\\;$[[0]]</p>\n      <p>Input numbers as fractions or integers and not as decimals.</p>\n      <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n      ", "scripts": {}, "useCustomName": false, "stepsPenalty": 0, "showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true}], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}