// Numbas version: exam_results_page_options
{"name": "Quotient rule - differentiate linear over quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "det": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a*f-b*d", "description": "", "name": "det"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..8)", "description": "", "name": "c1"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)", "description": "", "name": "d"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "description": "", "name": "f"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "a"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}}, "ungrouped_variables": ["a", "c", "b", "d", "f", "s2", "s1", "det", "c1"], "name": "Quotient rule - differentiate linear over quadratic", "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "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>", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "variableReplacements": []}], "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x^2+{d}x+{f})}\\]<br/>You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d}x+{f})^2}\\]<br/>for a polynomial $g(x)$.&nbsp;You are asked to find $g(x)$</p>\n\t\t\t        <p>$g(x)=\\;$[[0]]</p>\n\t\t\t        <p>Input numbers as fractions or integers and not as decimals.</p>\n\t\t\t        <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n\t\t\t", "stepsPenalty": 0, "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "answerSimplification": "std", "marks": 3, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["algebraic manipulation", "Calculus", "calculus", "checked2015", "derivative of a quotient", "Differentiation", "differentiation", "quotient rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>The derivative of $\\displaystyle \\frac{ax+b}{cx^2+dx+f}$ is $\\displaystyle \\frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.</p>"}, "advice": "<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<p>For this example:</p>\n<p>\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]</p>\n<p>\\[\\simplify[std]{v = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x+{d}}\\]</p>\n<p>Hence on substituting into the quotient rule above we get:</p>\n<p>\\[\\begin{eqnarray*} \\frac{df}{dx}&amp;=&amp;\\simplify[std]{({a}({c}x^2+{d}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\ &amp;=&amp;\\simplify[std]{({a*c}x^2+{a*d}x+{a*f}-{2*c*a}x^2-{a*d+2*c*b}x-{d*b})/({c}x^2+{d}x+{f})^2}\\\\ &amp;=&amp;\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2} \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}