// Numbas version: finer_feedback_settings
{"showQuestionGroupNames": false, "name": "Maria's copy of Differentiation using the quotient rule", "percentPass": 0, "type": "exam", "shuffleQuestions": false, "pickQuestions": 0, "allQuestions": true, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [{"name": "Differentiation: quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(2|a,random(-7..7#2),random(-8..8#2))", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,3,5,7)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d1=b*c,abs(d1)+1,d1)", "description": "", "name": "d"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..8)", "description": "", "name": "d1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "dPoly", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n      <p>\\[\\simplify[dPoly]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]</p>\n      <p>You are given that \\[\\simplify[dPoly]{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>You can click on Steps to get help. You will lose 1 mark if you do so.</p>\n      <p>$g(x)=\\;$[[0]]</p>\n      ", "steps": [{"type": "information", "prompt": "<p>The quotient rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule or otherwise.</p>", "tags": ["calculus", "Calculus", "checked2015", "derivatives", "derivatives ", "deriving functions", "differentiating a quotient", "differentiation", "mas1601", "MAS1601", "quotient rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                            \t\t<p><strong>20/06/2012:</strong></p>\n                            \t\t<p>Added tags.</p>\n                            \t\t<p>Feedback on entering and submitting a maths expression talks about being numerically correct. Perhaps some other wording here?</p>\n                            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $f(x) = (a x + b)/ \\sqrt{c x + d}$ and find $g(x)$ such that $ f^{\\prime}(x) = g(x)/ (2(c x + d)^{3/2})$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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{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[dPoly]{u = {a} * x + {b}}\\Rightarrow \\simplify{Diff(u,x,1) = {a}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[dPoly]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[dPoly]{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[dPoly]{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[dPoly]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].</p>\n  \n  \n  "}, {"name": "Differentiation: Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "name": "c1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "name": "c"}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d-b*c", "description": "", "name": "det"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "d"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "{det}/({c}x+{d})^2", "showCorrectAnswer": true, "vsetrange": [10, 11], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}*x+{d})}\\]</p>\n\t\t\t      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n\t\t\t", "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>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["Calculus", "calculus", "checked2015", "derivative of a quotient", "derivatives", "derivatives ", "differentiate a rational polynomial", "differentiation", "mas1601", "MAS1601", "quotient rule"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "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>Improved display of prompt.</p>\n\t\t<p>Checked calculation. OK.</p>\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate $\\displaystyle \\frac{ax+b}{cx+d}$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c}}\\]</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+{d})-{c}({a}x+{b}))/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{({a*c}x+{a*d}-{c*a}x-{c*b})/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{{det}/({c}x+{d})^2}\n\t  \n\t  \\end{eqnarray*}\\]</p>\n\t  \n\t  \n\t"}, {"name": "Differentiation: Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "name": "c1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "name": "c"}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d-b*c", "description": "", "name": "det"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "d"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "{-c*a}x^2+{-2*b*c}x+{a*d}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n        <p>\\[\\simplify[std]{f(x) = ({a} * x+{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 all 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        ", "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>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["algebraic manipulation", "calculus", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "mas1601", "quotient rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "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>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 input as fractions or integers, so added decimal point ot forbidden strings.</p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>The derivative of $\\displaystyle \\frac{ax+b}{cx^2+d}$ is of the form $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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})} \\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]{({a}({c}x^2+{d})-{2*c}x({a}x+{b}))/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({a*c}x^2+{a*d}-{2*c*a}x^2-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n  \n  &#38;=&#38;\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*d})/({c}x^2+{d})^2}\n  \n  \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*d}}$</p>\n  \n  \n    "}, {"name": "Quotient rule - differentiate exponential over exponential", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)", "name": "c", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..8)", "name": "b1", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a^2=b1^2,b1+1,b1)", "name": "b", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "s2", "s1", "b1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "{c*(b^2-a^2)}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a}+{b}e^({c}x))/({b}+{a}e^({c}x))}\\]</p>\n\t\t\t      <p>You are given that \\[\\simplify[std]{Diff(f,x,1) = (a*e^({c}x)) / ({b}+{a}e^({c}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      <p>You can click on Show steps to get help. You will not lose any marks if you do so.</p>\n\t\t\t", "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>", "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "quotient rule", "Steps"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "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 expressions don't change order from that intended. Got rid of a redundant ruleset.</p>\n\t\t<p>&nbsp;</p>\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>The derivative of $\\displaystyle \\frac{a+be^{cx}}{b+ae^{cx}}$ is $\\displaystyle \\frac{pe^{cx}} {(b+ae^{cx})^2}$. Find $p$.</p>"}, "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}+{b}e^({c}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b*c}e^({c}x)}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\simplify[std]{v = {b}+{a}e^({c}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*c}e^({c}x)}\\]</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]{({b*c}e^({c}x)({b}+{a}e^({c}x))-{a*c}e^({c}x)({a}+{b}e^({c}x)))/({b}+{a}e^({c}x))^2}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{({b^2*c} e^({c}x)+{a*b*c}*e^({2*c}x)-{a^2*c}e^({c}x)-{a*b*c}*e^({2*c}x) )/({b}+{a}e^({c}x))^2}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{({b^2*c} e^({c}x)-{a^2*c}e^({c}x))/({b}+{a}e^({c}x))^2}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{({b^2*c-a^2*c} e^({c}x))/({b}+{a}e^({c}x))^2}\t\n\t    \n\t    \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t    \n\t    <p>Hence $a=\\var{c*(b^2-a^2)}$</p>\n\t    \n\t    \n\t"}, {"name": "Quotient rule - differentiate linear over quadratic", "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/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "det": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a*f-b*d", "description": "", "name": "det"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..8)", "description": "", "name": "c1"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..9)", "description": "", "name": "d"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "description": "", "name": "f"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "a"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}}, "ungrouped_variables": ["a", "c", "b", "d", "f", "s2", "s1", "det", "c1"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "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>", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "variableReplacements": []}], "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", "stepsPenalty": 0, "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{-c*a}x^2+{-2*b*c}x+{a*f-b*d}", "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "answerSimplification": "std", "marks": 3, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["algebraic manipulation", "Calculus", "calculus", "checked2015", "derivative of a quotient", "Differentiation", "differentiation", "quotient rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "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>For this example:</p>\n<p>\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]</p>\n<p>\\[\\simplify[std]{v = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x+{d}}\\]</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}({c}x^2+{d}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\ &amp;=&amp;\\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}\\\\ &amp;=&amp;\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2} \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$</p>"}, {"name": "Quotient rule - differentiate linear over square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "name": "a", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d1=b*c,abs(d1)+1,d1)", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,3,5,7)", "name": "c", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(2|a,random(-7..7#2),random(-8..8#2))", "name": "b", "description": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..8)", "name": "d1", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "d1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as fractions or integers.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "all", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "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}], "prompt": "\n\t\t\t<p>\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]</p>\n\t\t\t      <p>You are given that \\[\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]</p>\n\t\t\t      <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n\t\t\t      <p>Input all numbers as fractions or integers.</p>\n\t\t\t      <p>You can click on Show steps to get help. You will not lose any marks if you do so.</p>\n\t\t\t      <p>$g(x)=\\;$[[0]]</p>\n\t\t\t", "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule or otherwise.</p>", "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "quotient rule", "Steps"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t    \n\t    \n\t    <p>The quotient rule says that if $u$ and $v$ are functions of $x$ then</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\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 = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({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>\\[\\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\t    \n\t    \n\t    \n\t    <p>Hence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].</p>\n\t    \n\t    \n\t"}, {"name": "Quotient rule - differentiate quadratic over quadratic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "name": "c1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "name": "c", "description": ""}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d-b*c", "name": "det", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "{2*det}x", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "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}], "prompt": "\n      <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      ", "marks": 0}], "statement": "<p>Differentiate the following function $f(x)$ using the quotient rule.</p>", "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "quotient rule", "Steps"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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  "}, {"name": "Quotient rule - differentiate quotient of linear terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..8)", "name": "c1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "name": "c", "description": ""}, "det": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d-b*c", "name": "det", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "{a}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-det}/{c}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<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\t\t\t        <p>$a=\\;$[[0]]</p>\n\t\t\t        <p>$b=\\;$[[1]]</p>\n\t\t\t        <p>You can click on Show steps to get some help, but you will lose 1 mark if you do so.</p>\n\t\t\t", "steps": [{"type": "information", "prompt": "<p>$\\simplify[std]{{a}x+{b}=a*({c}x+{d})+b}$ for suitable numbers $a$ and $b$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-c}/({c}x+{d})^2", "showCorrectAnswer": true, "vsetrange": [10, 11], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{det}/({c}x+{d})^2", "showCorrectAnswer": true, "vsetrange": [10, 11], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t<p>Differentiate<br/>\\[\\simplify[std]{g(x) = 1/({c}x+{d})}\\]</p>\n\t\t\t        <p>$\\displaystyle \\frac{dg}{dx}=\\;$[[0]]</p>\n\t\t\t        <p>Hence using the first part of the question differentiate \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>\n\t\t\t        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[1]]</p>\n\t\t\t        <p>Input numbers as fractions or integers and not as decimals.</p>\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Let \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]</p>", "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivatives", "derivatives ", "deriving a quotient", "differentiate a quotient", "differentiation", "dividing linear polynomials", "MAS1601"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "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>"}, "advice": "\n\t<p><strong>a)</strong></p>\n\t    <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\t    <p><strong>b)</strong></p>\n\t    <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,!unitPower,!unitDenominator,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\\\ &amp;=&amp;\\simplify[std]{{det}/({c}x+{d})^2} \\end{eqnarray*}\\]</p>\n\t"}, {"name": "Quotient rule - differentiate quotient of trig functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["checked2015"], "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"}, "statement": "<p>Differentiate the following functions using the quotient rule.</p>", "advice": "<p>The quotient rule says that if $u$ and $v$ are functions of $x$ then</p>\n<p>\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]</p>\n<h3>a)</h3>\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{align}<br/> \\frac{\\mathrm{d}f}{\\mathrm{d}x} &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} \\\\[0.5em]<br/> &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} \\\\[0.5em] <br/>&amp;= \\simplify[std]{({a*c}cos({a}x)^2+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] <br/>&amp;= \\simplify[std]{({a*c}(cos({a}x)^2+sin({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em]<br/> &amp;= \\simplify[std]{({a*c})/({b}sin({a}x)+{c}cos({a}x))^2} <br/>\\end{align}</p>\n<p>Hence $a=\\var{a*c}$.</p>\n<h3>b)</h3>\n<p>\\[\\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{align}<br/>\\frac{\\mathrm{d}g}{\\mathrm{d}x} &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} \\\\[0.5em]<br/> &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} \\\\[0.5em]<br/> &amp;= \\simplify[std]{({-a*b}sin({a}x)^2-{a*b}cos({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] <br/>&amp;= \\simplify[std]{({-a*b}(sin({a}x)^2+cos({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em]<br/> &amp;= \\simplify[std]{({-a*b})/({b}sin({a}x)+{c}cos({a}x))^2}<br/>\\end{align}</p>\n<p>Hence $b=\\var{-a*b}$.</p>\n<p><strong>c)</strong></p>\n<p>We have that $h(x)=\\simplify[std]{{m}f(x)+{n}g(x)}$.</p>\n<p>Hence</p>\n<p>\\begin{align}<br/>\\frac{\\mathrm{d}h}{\\mathrm{d}x} &amp;= \\simplify[std]{{m}*Diff(f,x,1)+{n}*Diff(f,x,1)} \\\\[0.5em] <br/>&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)} \\\\[0.5em] <br/>&amp;= \\simplify[std]{(({m}*{a*c})+({n}*{-a*b}))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] <br/>&amp;= \\simplify[std]{{res}/({b}sin({a}x)+{c}cos({a}x))^2} <br/>\\end{align}</p>\n<p>Hence $c=\\var{res}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noleadingMinus"]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "s1*random(2..8)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(m*c=n1*b,n1+1,n1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "s2*random(2..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "res": {"name": "res", "group": "Ungrouped variables", "definition": "m*a*c-n*b*a", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "if(b^2=c1^2,c1+1,c1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s2", "s1", "m", "n", "res", "n1", "c1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<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>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]</p>\n<p>You are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]</p>\n<p>for a number $b$. You have to find $b$.</p>\n<p>$b=$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-b*a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "9", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{res}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}], "pickQuestions": 0}], "feedback": {"advicethreshold": 0, "showactualmark": true, "showanswerstate": true, "showtotalmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Use the quotient rule to differentiate various functions.</p>"}, "navigation": {"allowregen": true, "showfrontpage": true, "browse": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": true, "reverse": true}, "questions": [], "duration": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": []}