// 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<br/>\\[\\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 = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*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>\\[\\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  \n  &#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  \n  &#38;=&#38;\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2}\n  \n  \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <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      <p>$g(x)=\\;$[[0]]</p>\n      <p>Input numbers as fractions or integers and not as decimals.</p>\n      <p>Click on Show steps for more information. You will not lose any marks by doing so.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "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.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "if(a*d=b*c1,c1+1,c1)", "name": "c"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "d": {"definition": "s2*random(1..9)", "name": "d"}, "f": {"definition": "random(-9..9)", "name": "f"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "det": {"definition": "a*f-b*d", "name": "det"}, "c1": {"definition": "random(1..8)", "name": "c1"}}, "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 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>", "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/"}]}