// Numbas version: exam_results_page_options
{"name": "Quotient rule - differentiate linear over square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quotient rule - differentiate linear over square root", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>The derivative of &nbsp;$\\displaystyle \\frac{ax+b}{\\sqrt{cx+d}}$ is $\\displaystyle \\frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.</p>"}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule or otherwise.</p>", "variable_groups": [], "functions": {}, "variables": {"b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "if(2|a,random(-7..7#2),random(-8..8#2))"}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1,3,5,7)"}, "d1": {"name": "d1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "s1*random(1..8)"}, "s1": {"name": "s1", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(1..8)"}, "d": {"name": "d", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "if(a*d1=b*c,abs(d1)+1,d1)"}}, "tags": [], "parts": [{"variableReplacements": [], "stepsPenalty": 0, "type": "gapfill", "useCustomName": false, "steps": [{"variableReplacements": [], "type": "information", "useCustomName": false, "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>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]</p>\n\t\t\t      <p>You are given that \\[\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]</p>\n\t\t\t      <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n\t\t\t      <p>Input all numbers as fractions or integers.</p>\n\t\t\t      <p>You can click on Show steps to get help. You will not lose any marks if you do so.</p>\n\t\t\t      <p>$g(x)=\\;$[[0]]</p>\n\t\t\t", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": false, "strings": ["."], "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers.</p>"}, "answerSimplification": "all", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 3, "vsetRangePoints": 5, "valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t    \n\t    \n\t    <p>The quotient rule says that if $u$ and $v$ are functions of $x$ then</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\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 = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\\]</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>\\[\\simplify[std]{Diff(f,x,1) = ({a} * Sqrt({c} * x + {d}) -(({a} * x + {b}) * Diff(v,x,1))) / ({c} * x + {d}) = ({a} * Sqrt({c} * x + {d}) -(({c} * ({a} * x + {b})) / (2 * Sqrt({c} * x + {d})))) / ({c} * x + {d}) = ({2 * a} * ({c} * x + {d}) -({c} * ({a} * x + {b}))) / (2 * ({c} * x + {d}) ^ (3 / 2)) = ({a * c} * x + {2 * a * d -(c * b)}) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>Hence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].</p>\n\t    \n\t    \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "d1"], "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/"}]}