\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}*x+{d})}\\]
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\n\t\t\t", "steps": [{"type": "information", "prompt": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
Differentiate the following function $f(x)$ using the quotient rule.
", "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\t1/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tImproved display of prompt.
\n\t\tChecked calculation. OK.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Differentiate $\\displaystyle \\frac{ax+b}{cx+d}$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tThe quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = ({c} * x+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c}}\\]
\n\t \n\t \n\t \n\tHence on substituting into the quotient rule above we get:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x+{d})-{c}({a}x+{b}))/({c}x+{d})^2}\\\\\n\t \n\t &=&\\simplify[std]{({a*c}x+{a*d}-{c*a}x-{c*b})/({c}x+{d})^2}\\\\\n\t \n\t &=&\\simplify[std]{{det}/({c}x+{d})^2}\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t"}, {"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\\[\\simplify[std]{f(x) = ({a}+{b}e^({c}x))/({b}+{a}e^({c}x))}\\]
\n\t\t\tYou are given that \\[\\simplify[std]{Diff(f,x,1) = (a*e^({c}x)) / ({b}+{a}e^({c}x))^2}\\]
\n\t\t\tfor a number $a$. You have to find $a$.
\n\t\t\t$a=\\;$[[0]]
\n\t\t\tYou can click on Show steps to get help. You will not lose any marks if you do so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
Differentiate the following function $f(x)$ using the quotient rule.
", "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\t1/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n\t\tChanged std rule set to include !noLeadingMinus, so expressions don't change order from that intended. Got rid of a redundant ruleset.
\n\t\t\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
The derivative of $\\displaystyle \\frac{a+be^{cx}}{b+ae^{cx}}$ is $\\displaystyle \\frac{pe^{cx}} {(b+ae^{cx})^2}$. Find $p$.
"}, "advice": "\n\t \n\t \n\tThe quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = {a}+{b}e^({c}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b*c}e^({c}x)}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = {b}+{a}e^({c}x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a*c}e^({c}x)}\\]
\n\t \n\t \n\t \n\tHence on substituting into the quotient rule above we get:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\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 &=&\\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 &=&\\simplify[std]{({b^2*c} e^({c}x)-{a^2*c}e^({c}x))/({b}+{a}e^({c}x))^2}\\\\\n\t \n\t &=&\\simplify[std]{({b^2*c-a^2*c} e^({c}x))/({b}+{a}e^({c}x))^2}\t\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t \n\tHence $a=\\var{c*(b^2-a^2)}$
\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": "Input all numbers as fractions or integers and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x^2+{d})}\\]
You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d})^2}\\]
for a polynomial $g(x)$. You are asked to find $g(x)$
$g(x)=\\;$[[0]]
\nInput all numbers as fractions or integers and not as decimals.
\nClick on Show steps for more information. You will not lose any marks by doing so.
\n ", "steps": [{"type": "information", "prompt": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
Differentiate the following function $f(x)$ using the quotient rule.
", "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\t1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n \t\tChanged std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.
\n \t\tImproved display in various places.
\n \t\tAdded condition that numbers input as fractions or integers, so added decimal point ot forbidden strings.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "The derivative of $\\displaystyle \\frac{ax+b}{cx^2+d}$ is of the form $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nThe quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
For this example:
\n \n \n \n\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n \n \n \n\\[\\simplify[std]{v = ({c} * x^2+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x}\\]
\n \n \n \nHence on substituting into the quotient rule above we get:
\n \n \n \n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x^2+{d})-{2*c}x({a}x+{b}))/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({a*c}x^2+{a*d}-{2*c*a}x^2-{2*c*b}x)/({c}x^2+{d})^2}\\\\\n \n &=&\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*d})/({c}x^2+{d})^2}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*d}}$
The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
\\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x^2+{d}x+{f})}\\]
You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d}x+{f})^2}\\]
for a polynomial $g(x)$. You are asked to find $g(x)$
$g(x)=\\;$[[0]]
\n\t\t\tInput numbers as fractions or integers and not as decimals.
\n\t\t\tClick on Show steps for more information. You will not lose any marks by doing so.
\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": "Input numbers as fractions or integers and not as decimals.
", "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": "Differentiate the following function $f(x)$ using the quotient rule.
", "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": "The derivative of $\\displaystyle \\frac{ax+b}{cx^2+dx+f}$ is $\\displaystyle \\frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.
"}, "advice": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
For this example:
\n\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n\\[\\simplify[std]{v = ({c} * x^2+{d}x+{f})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x+{d}}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({a}({c}x^2+{d}x+{f})-({2*c}x+{d})({a}x+{b}))/({c}x^2+{d}x+{f})^2}\\\\ &=&\\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}\\\\ &=&\\simplify[std]{({-c*a}x^2+{-2*b*c}x+{a*f-d*b})/({c}x^2+{d}x+{f})^2} \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$
Input all numbers as fractions or integers.
", "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": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]
\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]
\n\t\t\tYou are given that \\[\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]
\n\t\t\tfor a polynomial $g(x)$. You have to find $g(x)$.
\n\t\t\tInput all numbers as fractions or integers.
\n\t\t\tYou can click on Show steps to get help. You will not lose any marks if you do so.
\n\t\t\t$g(x)=\\;$[[0]]
\n\t\t\t", "marks": 0}], "statement": "Differentiate the following function $f(x)$ using the quotient rule or otherwise.
", "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\t1/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n\t\tChanged std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.
\n\t\tImproved display in various places.
\n\t\tAdded condition that numbers have to be input as fractions or integers - added decimal point to forbidden strings.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "The derivative of $\\displaystyle \\frac{ax+b}{\\sqrt{cx+d}}$ is $\\displaystyle \\frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tThe quotient rule says that if $u$ and $v$ are functions of $x$ then
\n\t \n\t \n\t \n\t\\[\\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]
\n\t \n\t \n\t \n\tFor this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = {a} * x + {b}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\\]
\n\t \n\t \n\t \n\tHence on substituting into the quotient rule above we get:
\n\t \n\t \n\t \n\t\\[\\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))}\\]
\n\t \n\t \n\t \n\tHence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].
\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": "Input numbers as fractions or integers and not as decimals.
", "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": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
\\[\\simplify[std]{f(x) = ({a} * x^2+{b})/({c}x^2+{d})}\\]
You are given that \\[\\frac{df}{dx}=\\simplify[std]{g(x)/({c}x^2+{d})^2}\\]
for a polynomial $g(x)$. You are asked to find $g(x)$
$g(x)=\\;$[[0]]
\nInput numbers as fractions or integers and not as decimals.
\nClick on Show steps for more information. You will not lose any marks by doing so.
\n ", "marks": 0}], "statement": "Differentiate the following function $f(x)$ using the quotient rule.
", "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\t1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n \t\tChanged std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.
\n \t\tImproved display in various places.
\n \t\tAdded condition that numbers have to be inout as fractions or integers - added decimal point to forbidden strings.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "The derivative of $\\displaystyle \\frac{ax^2+b}{cx^2+d}$ is $\\displaystyle \\frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nThe quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
For this example:
\n \n \n \n\\[\\simplify[std]{u = ({a}x^2+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {2*a}x}\\]
\n \n \n \n\\[\\simplify[std]{v = ({c} * x^2+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*c}x}\\]
\n \n \n \nHence on substituting into the quotient rule above we get:
\n \n \n \n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\simplify[std]{({2*a}x({c}x^2+{d})-{2*c}x({a}x^2+{b}))/({c}x^2+{d})^2}\\\\\n \n &=&\\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 &=&\\simplify[std]{({2*det}x)/({c}x^2+{d})^2}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{2*det}x}$
Input numbers as fractions or integers and not as decimals.
", "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": "Input numbers as fractions or integers and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\tFind numbers $a$ and $b$ such that
\\[\\simplify[std]{f(x) = a + b/({c}x+{d})}\\]
Enter a and b as integers or fractions, but not as decimals.
$a=\\;$[[0]]
\n\t\t\t$b=\\;$[[1]]
\n\t\t\tYou can click on Show steps to get some help, but you will lose 1 mark if you do so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "$\\simplify[std]{{a}x+{b}=a*({c}x+{d})+b}$ for suitable numbers $a$ and $b$.
", "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": "Input numbers as fractions or integers and not as decimals.
", "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": "Input numbers as fractions or integers and not as decimals.
", "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\tDifferentiate
\\[\\simplify[std]{g(x) = 1/({c}x+{d})}\\]
$\\displaystyle \\frac{dg}{dx}=\\;$[[0]]
\n\t\t\tHence using the first part of the question differentiate \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[1]]
\n\t\t\tInput numbers as fractions or integers and not as decimals.
\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "Let \\[\\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}\\]
", "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\t1/08/2012:
\n\t\tAdded tags.
\n\t\tChecked calculation. OK.
\n\t\tAdded description.
\n\t\tAll round improvement in display.
\n\t\tAdded forbidden instructions on using decimals.
\n\t\tAdded information on losing 1 mark if use Show steps in part a).
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Other method. Find $p,\\;q$ such that $\\displaystyle \\frac{ax+b}{cx+d}= p+ \\frac{q}{cx+d}$. Find the derivative of $\\displaystyle \\frac{ax+b}{cx+d}$.
"}, "advice": "\n\ta)
\n\tWe have $\\displaystyle \\simplify[std]{{a}x+{b}={a}/{c}*({c}x+{d})+{b}-{a}*{d}/{c}={a}/{c}*({c}x+{d})+{-det}/{c}}$
Hence \\[\\begin{eqnarray*} \\simplify[std]{({a} * x+{b})/({c}x+{d})}&=&\\simplify[std]{({a}/{c}*({c}x+{d})+{-det}/{c})/({c}x+{d})}\\\\ &=&\\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})} \\end{eqnarray*}\\]
Where we have divided out by $\\simplify[std]{{c}x+{d}}$ at the last step.
b)
\n\tWe have \\[\\frac{dg}{dx} = \\simplify[std]{{-c}/({c}x+{d})^2}\\]
using standard rules of differentiation.
Since from a), \\[f(x) = \\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})}\\]
we see that
\\[\\begin{eqnarray*}\\frac{df}{dx} &=&\\simplify[std,!unitPower,!unitDenominator,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\\\ &=&\\simplify[std]{{det}/({c}x+{d})^2} \\end{eqnarray*}\\]
Find $\\displaystyle \\frac{d}{dx}\\left(\\frac{m\\sin(ax)+n\\cos(ax)}{b\\sin(ax)+c\\cos(ax)}\\right)$. Three part question.
", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]
a)
\nFor this example:
\n\\[\\simplify[std]{u = sin({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}cos({a}x)}\\]
\n\\[\\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)}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{df}{dx}&=&\\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}\\\\ &=&\\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}\\\\ &=&\\simplify[std]{({a*c}cos({a}x)^2+{a*c}sin({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*c}(cos({a}x)^2+sin({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({a*c})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
\nHence $a=\\var{a*c}$
\nb)\\[\\simplify[std]{u = cos({a}x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = -{a}sin({a}x)}\\]
\n\\[\\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)}\\]
\nHence on substituting into the quotient rule above we get:
\n\\[\\begin{eqnarray*} \\frac{dg}{dx}&=&\\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}\\\\ &=&\\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}\\\\ &=&\\simplify[std]{({-a*b}sin({a}x)^2-{a*b}cos({a}x)^2)/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b}(sin({a}x)^2+cos({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{({-a*b})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
\nHence $b=\\var{-a*b}$
\nc)
\nWe have that $h(x)=\\simplify[std]{{m}f(x)+{n}g(x)}$
Hence \\[\\begin{eqnarray*}\\frac{dh}{dx} &=& \\simplify[std]{{m}*Diff(f,x,1)+{n}*Diff(g,x,1)}\\\\ &=&\\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)}\\\\ &=&\\simplify[std]{(({m}*{a*c})+({n}*{-a*b}))/({b}sin({a}x)+{c}cos({a}x))^2}\\\\ &=&\\simplify[std]{{res}/({b}sin({a}x)+{c}cos({a}x))^2} \\end{eqnarray*}\\]
Hence $c=\\var{res}$
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\n\t\t\tYou are given that \\[\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\tfor a number $a$. You have to find $a$.
\n\t\t\t$a=\\;$[[0]]
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\\[\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\\]
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\tYou are given that \\[\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\tfor a number $b$. You have to find $b$.
\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t$b=\\;$[[0]]
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\n\t\t\tYou are given that \\[\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\\]
\n\t\t\tfor a number $c$. You have to find $c$.
\n\t\t\t$c=\\;$[[0]]
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