// Numbas version: finer_feedback_settings
{"name": "Luis's copy of 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": [{"ungrouped_variables": ["a", "c", "b", "d", "f", "s2", "s1", "det", "c1"], "name": "Luis's copy of Quotient rule - differentiate linear over quadratic", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "parts": [{"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", "gaps": [{"checkvariablenames": false, "showpreview": true, "vsetrangepoints": 5, "answersimplification": "std", "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 3, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "strings": ["."], "showStrings": false, "partialCredit": 0}}], "steps": [{"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>", "scripts": {}, "marks": 0, "showCorrectAnswer": true}], "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "stepsPenalty": 0, "marks": 0}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "functions": {}, "variables": {"c1": {"name": "c1", "templateType": "anything", "description": "", "definition": "random(1..8)", "group": "Ungrouped variables"}, "d": {"name": "d", "templateType": "anything", "description": "", "definition": "s2*random(1..9)", "group": "Ungrouped variables"}, "s1": {"name": "s1", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "det": {"name": "det", "templateType": "anything", "description": "", "definition": "a*f-b*d", "group": "Ungrouped variables"}, "b": {"name": "b", "templateType": "anything", "description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables"}, "s2": {"name": "s2", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "c": {"name": "c", "templateType": "anything", "description": "", "definition": "if(a*d=b*c1,c1+1,c1)", "group": "Ungrouped variables"}, "a": {"name": "a", "templateType": "anything", "description": "", "definition": "random(2..9)", "group": "Ungrouped variables"}, "f": {"name": "f", "templateType": "anything", "description": "", "definition": "random(-9..9)", "group": "Ungrouped variables"}}, "variable_groups": [], "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "quotient rule", "Steps"], "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^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*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>\\[\\begin{eqnarray*} \\frac{df}{dx}&#38;=&#38;\\simplify[std]{({a}({c}x^2+{d}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\\n\t    \n\t    &#38;=&#38;\\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}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2}\n\t    \n\t    \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$</p>\n\t    \n\t    \n\t", "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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 inout as fractions or integers - added decimal point to forbidden strings.</p>\n\t\t<p>&nbsp;</p>\n\t\t", "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>"}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}