// Numbas version: exam_results_page_options
{"name": "Differentiation: Quotient rule [L8]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "extensions": [], "parts": [{"prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}*x+{d})}\\]</p>\n\t\t\t      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n\t\t\t", "marks": 0, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "unitTests": [], "sortAnswers": false, "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "stepsPenalty": 1, "gaps": [{"showPreview": true, "marks": 3, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "unitTests": [], "checkingAccuracy": 0.001, "type": "jme", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "failureRate": 1, "variableReplacementStrategy": "originalfirst", "answer": "{det}/({c}x+{d})^2", "answerSimplification": "std", "vsetRange": [10, 11], "expectedVariableNames": [], "vsetRangePoints": 5, "checkVariableNames": false, "showCorrectAnswer": true, "checkingType": "absdiff"}], "showCorrectAnswer": true, "steps": [{"unitTests": [], "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>", "type": "information", "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacements": []}]}], "advice": "\n\t  \n\t  \n\t  <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\t  \n\t  \n\t  \n\t  <p>For this example:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{v = ({c} * x+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c}}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>Hence on substituting into the quotient rule above we get:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\begin{eqnarray*} \\frac{df}{dx}&#38;=&#38;\\simplify[std]{({a}({c}x+{d})-{c}({a}x+{b}))/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{({a*c}x+{a*d}-{c*a}x-{c*b})/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{{det}/({c}x+{d})^2}\n\t  \n\t  \\end{eqnarray*}\\]</p>\n\t  \n\t  \n\t", "preamble": {"css": "", "js": ""}, "variables": {"det": {"definition": "a*d-b*c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "det"}, "c": {"definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c"}, "a": {"definition": "random(2..9)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "a"}, "s2": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s2"}, "s1": {"definition": "random(1,-1)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "b"}, "d": {"definition": "s2*random(1..9)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "d"}, "c1": {"definition": "random(1..8)", "description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "c1"}}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "tags": ["L8"], "metadata": {"description": "<p>Differentiate $\\displaystyle \\frac{ax+b}{cx+d}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Differentiation: Quotient rule [L8]", "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "type": "question", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}]}