// Numbas version: exam_results_page_options
{"name": "Differentiation: quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(2|a,random(-7..7#2),random(-8..8#2))", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,3,5,7)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d1=b*c,abs(d1)+1,d1)", "description": "", "name": "d"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..8)", "description": "", "name": "d1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Differentiation: quotient rule", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "dPoly", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n      <p>\\[\\simplify[dPoly]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]</p>\n      <p>You are given that \\[\\simplify[dPoly]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]</p>\n      <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n      <p>You can click on Steps to get help. You will lose 1 mark if you do so.</p>\n      <p>$g(x)=\\;$[[0]]</p>\n      ", "steps": [{"type": "information", "prompt": "<p>The quotient rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule or otherwise.</p>", "tags": ["calculus", "Calculus", "checked2015", "derivatives", "derivatives ", "deriving functions", "differentiating a quotient", "differentiation", "mas1601", "MAS1601", "quotient rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                            \t\t<p><strong>20/06/2012:</strong></p>\n                            \t\t<p>Added tags.</p>\n                            \t\t<p>Feedback on entering and submitting a maths expression talks about being numerically correct. Perhaps some other wording here?</p>\n                            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $f(x) = (a x + b)/ \\sqrt{c x + d}$ and find $g(x)$ such that $ f^{\\prime}(x) = g(x)/ (2(c x + d)^{3/2})$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n  \n  \n  <p>The quotient rule says that if $u$ and $v$ are functions of $x$ then</p>\n  \n  \n  \n  <p>\\[\\simplify{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[dPoly]{u = {a} * x + {b}}\\Rightarrow \\simplify{Diff(u,x,1) = {a}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[dPoly]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[dPoly]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the quotient rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[dPoly]{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  \n  \n  \n  <p>Hence \\[\\simplify[dPoly]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].</p>\n  \n  \n  ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}