// Numbas version: exam_results_page_options
{"name": "Simon's copy of Quotient rule - differentiate quotient of trig functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "functions": {}, "metadata": {"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"}, "advice": "<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(g,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>", "statement": "<p>Differentiate the following functions using the quotient rule.</p>", "variable_groups": [], "name": "Simon's copy of Quotient rule - differentiate quotient of trig functions", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noleadingMinus"]}, "tags": [], "variables": {"res": {"name": "res", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "m*a*c-n*b*a"}, "s2": {"name": "s2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "n1": {"name": "n1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)"}, "b": {"name": "b", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)"}, "c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(b^2=c1^2,c1+1,c1)"}, "s1": {"name": "s1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "a": {"name": "a", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)"}, "n": {"name": "n", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(m*c=n1*b,n1+1,n1)"}, "c1": {"name": "c1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..8)"}, "m": {"name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)"}}, "parts": [{"prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = (sin({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n\t\t\t        <p>You are given that \\[\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n\t\t\t        <p>for a number $a$. You have to find $a$.</p>\n\t\t\t        <p>$a=\\;$[[0]]</p>\n\t\t\t", "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": 0, "customName": "", "useCustomName": false, "sortAnswers": false, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "answer": "{a*c}", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "marks": 3, "customName": "", "useCustomName": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "checkVariableNames": false}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "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>", "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": 0, "customName": "", "useCustomName": false, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "type": "information", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true}], "stepsPenalty": 0, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"prompt": "\n\t\t\t      \n\t\t\t      \n\t\t\t      <p>\\[\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n\t\t\t      \n\t\t\t      \n\t\t\t      \n\t\t\t      <p>You are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n\t\t\t      \n\t\t\t      \n\t\t\t      \n\t\t\t      <p>for a number $b$. You have to find $b$.</p>\n\t\t\t      \n\t\t\t      \n\t\t\t      \n\t\t\t      <p>$b=\\;$[[0]]</p>\n\t\t\t      \n\t\t\t      \n\t\t\t", "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": 0, "customName": "", "useCustomName": false, "sortAnswers": false, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "answer": "{-b*a}", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "marks": 3, "customName": "", "useCustomName": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "checkVariableNames": false}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"prompt": "\n\t\t\t<p>\\[\\simplify[std]{h(x) = ({m}sin({a}x)+{n}cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n\t\t\t        <p>You are given that \\[\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n\t\t\t        <p>for a number $c$. You have to find $c$.</p>\n\t\t\t        <p>$c=\\;$[[0]]</p>\n\t\t\t", "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": 0, "customName": "", "useCustomName": false, "sortAnswers": false, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "showPreview": true, "answer": "{res}", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "marks": 3, "customName": "", "useCustomName": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "checkVariableNames": false}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true}], "ungrouped_variables": ["a", "c", "b", "s2", "s1", "m", "n", "res", "n1", "c1"], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}