// Numbas version: finer_feedback_settings
{"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Quotient rule", "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivative of a quotient", "differentiation", "quotient rule", "steps"], "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[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 + {b}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{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[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  \n  \n  \n  <p>Hence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]</p>\n        <p>You are given that \\[\\simplify[std]{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>Input all numbers as fractions or integers.</p>\n        <p>You can click on Show steps to get help. You will not lose any marks if you do so.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 3.0, "answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "type": "jme"}], "steps": [{"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", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule or otherwise.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..8)", "name": "a"}, "c": {"definition": "random(1,3,5,7)", "name": "c"}, "b": {"definition": "if(2|a,random(-7..7#2),random(-8..8#2))", "name": "b"}, "d": {"definition": "if(a*d1=b*c,abs(d1)+1,d1)", "name": "d"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "d1": {"definition": "s1*random(1..8)", "name": "d1"}}, "metadata": {"notes": "\n    \t\t<p><strong>1/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Added information about Show steps. Altered to 0 marks lost rather than 1.</p>\n    \t\t<p>Changed std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.</p>\n    \t\t<p>Improved display in various places.</p>\n    \t\t<p>Added condition that numbers have to be input as fractions or integers - added decimal point to forbidden strings.</p>\n    \t\t", "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>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}