// 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", "algebraic manipulation", "calculus", "derivatives", "deriving a quotient", "differentiate a quotient", "differentiation", "dividing linear polynomials"], "advice": "<p><strong>a)</strong></p>\n    <p>We have $\\displaystyle \\simplify[std]{{a}x+{b}={a}/{c}*({c}x+{d})+{b}-{a}*{d}/{c}={a}/{c}*({c}x+{d})+{-det}/{c}}$<br/>Hence \\[\\begin{eqnarray*} \\simplify[std]{({a} * x+{b})/({c}x+{d})}&amp;=&amp;\\simplify[std]{({a}/{c}*({c}x+{d})+{-det}/{c})/({c}x+{d})}\\\\ &amp;=&amp;\\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})} \\end{eqnarray*}\\]<br/>Where we have divided out by $\\simplify[std]{{c}x+{d}}$ at the last step.</p>\n    <p><strong>b)</strong></p>\n    <p>We have \\[\\frac{dg}{dx} = \\simplify[std]{{-c}/({c}x+{d})^2}\\]<br/>using standard rules of differentiation.<br/>Since from <strong>a)</strong>,&nbsp;\\[f(x) = \\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})}\\]<br/>&nbsp;we see that<br/>\\[\\begin{eqnarray*}\\frac{df}{dx} &amp;=&amp;\\simplify[std,fractionNumbers,!unitPower,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\\\ &amp;=&amp;\\simplify[std]{{det}/({c}x+{d})^2} \\end{eqnarray*}\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n        <p>Find numbers $a$ and $b$ such that<br/>\\[\\simplify[std]{f(x) = a + b/({c}x+{d})}\\]<br/>Enter a and b as integers or fractions, but not as decimals.</p>\n        <p>$a=\\;$[[0]]</p>\n        <p>$b=\\;$[[1]]</p>\n        <p>You can click on Show steps to get some help, but you will lose 1 mark if you do 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": 1.0, "answer": "{a}/{c}", "type": "jme"}, {"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": 1.0, "answer": "{-det}/{c}", "type": "jme"}], "steps": [{"prompt": "<p>$\\simplify[std]{{a}x+{b}=a*({c}x+{d})+b}$ for suitable numbers $a$ and $b$.</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n        <p>Differentiate<br/>\\[\\simplify[std]{g(x) = 1/({c}x+{d})}\\]</p>\n        <p>$\\displaystyle \\frac{dg}{dx}=\\;$[[0]]</p>\n        <p>Hence using the first part of the question differentiate \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[1]]</p>\n        <p>Input numbers as fractions or integers and not as decimals.</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": [10.0, 11.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{-c}/({c}x+{d})^2", "type": "jme"}, {"notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [10.0, 11.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{det}/({c}x+{d})^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "<p>Let \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</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"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "det": {"definition": "a*d-b*c", "name": "det"}, "c1": {"definition": "random(2..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>Checked calculation. OK.</p>\n        \t\t<p>Added description.</p>\n        \t\t<p>All round improvement in display.</p>\n        \t\t<p>Added &nbsp;forbidden instructions on using decimals.</p>\n        \t\t<p>Added information on losing 1 mark if use Show steps in part a).</p>\n        \t\t", "description": "<p>Other method. Find $p,\\;q$ such that&nbsp;$\\displaystyle \\frac{ax+b}{cx+d}= p+ \\frac{q}{cx+d}$. Find the derivative of $\\displaystyle \\frac{ax+b}{cx+d}$.</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/"}]}