// Numbas version: exam_results_page_options
{"name": "Maths Support: Quotient rule", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>8 questions on the quotient rule in differentiation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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}], "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": []}]}, {"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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"}], "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": []}]}, {"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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"}], "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": []}]}, {"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": [], "advice": "\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  <p><strong>a)</strong></p>\n  <p>For this example:</p>\n  <p>\\[\\simplify[std]{u = sin({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}cos({a}x)}\\]</p>\n  <p>\\[\\simplify[std]{v = {b}sin({a}x)+{c}cos({a}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*b}cos({a}x)+{-a*c}sin({a}x)}\\]</p>\n  <p>Hence on substituting into the quotient rule above we get:</p>\n  <p>\\[\\begin{eqnarray*} \\frac{df}{dx}&amp;=&amp;\\simplify[std]{({a}cos({a}x)({b}sin({a}x)+{c}cos({a}x))-sin({a}x)({a*b}cos({a}x)+{-a*c}sin({a}x)))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({a*b} cos({a}x) sin({a}x)+{a*c} cos({a}x)^2-{a*b} sin({a}x)cos({a}x)+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({a*c}cos({a}x)^2+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({a*c}(cos({a}x)^2+sin({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({a*c})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]</p>\n  <p>Hence $a=\\var{a*c}$</p>\n  <p><strong>b)</strong>\\[\\simplify[std]{u = cos({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = -{a}sin({a}x)}\\]</p>\n  <p>\\[\\simplify[std]{v = {b}sin({a}x)+{c}cos({a}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*b}cos({a}x)+{-a*c}sin({a}x)}\\]</p>\n  <p>Hence on substituting into the quotient rule above we get:</p>\n  <p>\\[\\begin{eqnarray*} \\frac{dg}{dx}&amp;=&amp;\\simplify[std]{({-a}sin({a}x)({b}sin({a}x)+{c}cos({a}x))-cos({a}x)({a*b}cos({a}x)+{-a*c}sin({a}x)))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({-a*b}sin({a}x)^2-{a*c} sin({a}x)cos({a}x)-{a*b}cos({a}x)^2+{a*c}sin({a})cos({a}x))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({-a*b}sin({a}x)^2-{a*b}cos({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({-a*b}(sin({a}x)^2+cos({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{({-a*b})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]</p>\n  <p>Hence $b=\\var{-a*b}$</p>\n  <p><strong>c)</strong></p>\n  <p>We have that $h(x)=\\simplify[std]{{m}f(x)+{n}g(x)}$<br/>Hence \\[\\begin{eqnarray*}\\frac{dh}{dx} &amp;=&amp; \\simplify[std]{{m}*Diff(f,x,1)+{n}*Diff(f,x,1)}\\\\ &amp;=&amp;\\simplify[std]{{m}*({a*c}/({b}sin({a}x)+{c}cos({a}x))^2)+{n}({-a*b}/({b}sin({a}x)+{c}cos({a}x))^2)}\\\\ &amp;=&amp;\\simplify[std]{(({m}*{a*c})+({n}*{-a*b}))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &amp;=&amp;\\simplify[std]{{res}/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]</p>\n  <p>Hence $c=\\var{res}$</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noleadingMinus"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>\\[\\simplify[std]{f(x) = (sin({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n      <p>You are given that \\[\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n      <p>for a number $a$. You have to find $a$.</p>\n      <p>$a=\\;$[[0]]</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{a*c}", "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"}, {"prompt": "\n        \n        \n        <p>\\[\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n        \n        \n        \n        <p>You are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n        \n        \n        \n        <p>for a number $b$. You have to find $b$.</p>\n        \n        \n        \n        <p>$b=\\;$[[0]]</p>\n        \n        \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{-b*a}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>\\[\\simplify[std]{h(x) = ({m}sin({a}x)+{n}cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n      <p>You are given that \\[\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n      <p>for a number $c$. You have to find $c$.</p>\n      <p>$c=\\;$[[0]]</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 3.0, "answer": "{res}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Differentiate the following functions using the quotient rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "if(b^2=c1^2,c1+1,c1)", "name": "c"}, "b": {"definition": "random(1..9)", "name": "b"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "random(2..9)", "name": "m"}, "n": {"definition": "if(m*c=n1*b,n1+1,n1)", "name": "n"}, "res": {"definition": "m*a*c-n*b*a", "name": "res"}, "n1": {"definition": "s2*random(2..9)", "name": "n1"}, "c1": {"definition": "s1*random(2..8)", "name": "c1"}}, "metadata": {"notes": "\n    \t\t<p>&nbsp;</p>\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>Changed std rule set to include !noLeadingMinus, so expressions 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>Changed to 0 penalty for accessing Show steps in first question.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\frac{d}{dx}\\left(\\frac{m\\sin(ax)+n\\cos(ax)}{b\\sin(ax)+c\\cos(ax)}\\right)$. Three part question.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Quotient rule. (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivative of a quotient", "differentiation", "quotient rule", "steps", "video"], "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^2+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {2*a}x}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({c} * x^2+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x}\\]</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]{({2*a}x({c}x^2+{d})-{2*c}x({a}x^2+{b}))/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({2*a*c}x^3+{2*a*d}x-{2*c*a}x^3-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({2*det}x)/({c}x^2+{d})^2}\n  \n  \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{2*det}x}$</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 0.0, "prompt": "<p>\\[\\simplify[std]{f(x) = ({a} * x^2+{b})/({c}x^2+{d})}\\]<br/>You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d})^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      <p>You will also find a video in Show steps showing how to apply the quotient rule.</p>", "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": "{2*det}x", "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>\n          <p><iframe src=\"https://player.vimeo.com/video/8177110\" frameborder=\"0\" width=\"640\" height=\"480\"></iframe></p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "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"}, "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(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", "description": "<p>The derivative of $\\displaystyle \\frac{ax^2+b}{cx^2+d}$ is $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.</p>\n            <p>Contains a video solving a similar quotient rule example. Although does not explicitly find $g(x)$ as asked in the question, but this is obvious.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}