// Numbas version: exam_results_page_options {"pickQuestions": 0, "shuffleQuestions": false, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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

\$\\simplify[dPoly]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\$

\n

You are given that \$\\simplify[dPoly]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\$

\n

for a polynomial $g(x)$. You have to find $g(x)$.

\n

You can click on Steps to get help. You will lose 1 mark if you do so.

\n

$g(x)=\\;$[[0]]

\n ", "steps": [{"type": "information", "prompt": "

The quotient rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\$

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

Differentiate the following function $f(x)$ using the quotient rule or otherwise.

", "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

20/06/2012:

\n \t\t

\n \t\t

Feedback on entering and submitting a maths expression talks about being numerically correct. Perhaps some other wording here?

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Differentiate $f(x) = (a x + b)/ \\sqrt{c x + d}$ and find $g(x)$ such that $f^{\\prime}(x) = g(x)/ (2(c x + d)^{3/2})$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \n

The quotient rule says that if $u$ and $v$ are functions of $x$ then

\n \n \n \n

\$\\simplify{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\$

\n \n \n \n

For this example:

\n \n \n \n

\$\\simplify[dPoly]{u = {a} * x + {b}}\\Rightarrow \\simplify{Diff(u,x,1) = {a}}\$

\n \n \n \n

\$\\simplify[dPoly]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[dPoly]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\$

\n \n \n \n

Hence on substituting into the quotient rule above we get:

\n \n \n \n

\$\\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))}\$

\n \n \n \n

Hence \$\\simplify[dPoly]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\$.

\n \n \n "}, {"name": "Differentiation: Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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

\$\\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}\$

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

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\t

1/08/2012:

\n\t\t

\n\t\t

\n\t\t

Improved display of prompt.

\n\t\t

Checked 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\t

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}\$

\n\t \n\t \n\t \n\t

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\t

Hence 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": "Differentiation: Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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.

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\$\\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 $g(x)$

\n

$g(x)=\\;$[[0]]

\n

Input all numbers as fractions or integers and not as decimals.

\n

Click 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}\$

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

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\t

1/08/2012:

\n \t\t

\n \t\t

\n \t\t

Added information about Show steps. Altered to 0 marks lost rather than 1.

\n \t\t

Changed std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.

\n \t\t

Improved display in various places.

\n \t\t

Added 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 \n

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}\$

\n \n \n \n

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 \n

Hence 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 $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*d}}$

\n \n \n "}, {"name": "Quotient rule - differentiate exponential over exponential", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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\t

You are given that \$\\simplify[std]{Diff(f,x,1) = (a*e^({c}x)) / ({b}+{a}e^({c}x))^2}\$

\n\t\t\t

for a number $a$. You have to find $a$.

\n\t\t\t

$a=\\;$[[0]]

\n\t\t\t

You 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}\$

", "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "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\t

1/08/2012:

\n\t\t

\n\t\t

\n\t\t

Checked calculation. OK.

\n\t\t

Added information about Show steps. Altered to 0 marks lost rather than 1.

\n\t\t

Changed 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 $p$.

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}\$

\n\t \n\t \n\t \n\t

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\t

Hence 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\t

Hence $a=\\var{c*(b^2-a^2)}$

\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}, "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": "

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}\$

", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "marks": 0, "variableReplacements": []}], "prompt": "\n\t\t\t

\$\\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 $g(x)$

\n\t\t\t

$g(x)=\\;$[[0]]

\n\t\t\t

Input numbers as fractions or integers and not as decimals.

\n\t\t\t

Click 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 $g(x)$.

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}\$

\n

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}}\$

\n

Hence 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 $g(x)=\\simplify[std]{{-c*a}x^2+{-2*b*c}x+{a*f-d*b}}$

"}, {"name": "Quotient rule - differentiate linear over square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "

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}\$

", "scripts": {}, "marks": 0}], "prompt": "\n\t\t\t

\$\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\$

\n\t\t\t

You are given that \$\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\$

\n\t\t\t

for a polynomial $g(x)$. You have to find $g(x)$.

\n\t\t\t

Input all numbers as fractions or integers.

\n\t\t\t

You 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\t

1/08/2012:

\n\t\t

\n\t\t

\n\t\t

Checked calculation. OK.

\n\t\t

Added information about Show steps. Altered to 0 marks lost rather than 1.

\n\t\t

Changed std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.

\n\t\t

Improved display in various places.

\n\t\t

Added 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\t

The 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\t

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 = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\$

\n\t \n\t \n\t \n\t

Hence 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\t

Hence \$\\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}, "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}\$

", "scripts": {}, "marks": 0}], "prompt": "\n

\$\\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 $g(x)$

\n

$g(x)=\\;$[[0]]

\n

Input numbers as fractions or integers and not as decimals.

\n

Click 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\t

1/08/2012:

\n \t\t

\n \t\t

\n \t\t

Checked calculation. OK.

\n \t\t

Added information about Show steps. Altered to 0 marks lost rather than 1.

\n \t\t

Changed std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.

\n \t\t

Improved display in various places.

\n \t\t

Added 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 \n

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}\$

\n \n \n \n

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 \n

Hence 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 $g(x)=\\simplify[std]{{2*det}x}$

\n \n \n "}, {"name": "Quotient rule - differentiate quotient of linear terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "

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\t

Find 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.

\n\t\t\t

$a=\\;$[[0]]

\n\t\t\t

$b=\\;$[[1]]

\n\t\t\t

You 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\t

Differentiate
\$\\simplify[std]{g(x) = 1/({c}x+{d})}\$

\n\t\t\t

$\\displaystyle \\frac{dg}{dx}=\\;$[[0]]

\n\t\t\t

Hence 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\t

Input 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\t

1/08/2012:

\n\t\t

\n\t\t

Checked calculation. OK.

\n\t\t

\n\t\t

All round improvement in display.

\n\t\t

Added  forbidden instructions on using decimals.

\n\t\t

Added 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 $\\displaystyle \\frac{ax+b}{cx+d}$.

a)

\n\t

We 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.

\n\t

b)

\n\t

We 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*}\$

\n\t"}, {"name": "Quotient rule - differentiate quotient of trig functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": 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": "

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"}, "statement": "

Differentiate the following functions using the quotient rule.

The quotient rule says that if $u$ and $v$ are functions of $x$ then

\n

\$\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\$

\n

### a)

\n

For 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)}\$

\n

Hence on substituting into the quotient rule above we get:

\n

\\begin{align}
\\frac{\\mathrm{d}f}{\\mathrm{d}x} &= \\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] &= \\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] &= \\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] &= \\simplify[std]{({a*c}(cos({a}x)^2+sin({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] &= \\simplify[std]{({a*c})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{align} \n Hence a=\\var{a*c}. \n ### b) \n \\[\\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)}\$

\n

Hence on substituting into the quotient rule above we get:

\n

\\begin{align}
\\frac{\\mathrm{d}g}{\\mathrm{d}x} &= \\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] &= \\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] &= \\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] &= \\simplify[std]{({-a*b}(sin({a}x)^2+cos({a}x)^2))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] &= \\simplify[std]{({-a*b})/({b}sin({a}x)+{c}cos({a}x))^2} \\end{align} \n Hence b=\\var{-a*b}. \n c) \n We have that h(x)=\\simplify[std]{{m}f(x)+{n}g(x)}. \n Hence \n \\begin{align} \\frac{\\mathrm{d}h}{\\mathrm{d}x} &= \\simplify[std]{{m}*Diff(f,x,1)+{n}*Diff(f,x,1)} \\\\[0.5em] &= \\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] &= \\simplify[std]{(({m}*{a*c})+({n}*{-a*b}))/({b}sin({a}x)+{c}cos({a}x))^2} \\\\[0.5em] &= \\simplify[std]{{res}/({b}sin({a}x)+{c}cos({a}x))^2} \\end{align} \n Hence c=\\var{res}. ", "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": " \\[\\simplify[std]{f(x) = (sin({a}x))/({b}sin({a}x)+{c}cos({a}x))}\

\n

You are given that \$\\simplify[std]{Diff(f,x,1) = a / ({b}sin({a}x)+{c}cos({a}x))^2}\$

\n

for a number $a$. You have to find $a$.

\n

$a=$ [[0]]

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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}\$

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\$\\simplify[std]{g(x) = (cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\$

\n

You are given that \$\\simplify[std]{Diff(g,x,1) = b / ({b}sin({a}x)+{c}cos({a}x))^2}\$

\n

for a number $b$. You have to find $b$.

\n

$b=$ [[0]]

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\$\\simplify[std]{h(x) = ({m}sin({a}x)+{n}cos({a}x))/({b}sin({a}x)+{c}cos({a}x))}\$

\n

You are given that \$\\simplify[std]{Diff(h,x,1) = c / ({b}sin({a}x)+{c}cos({a}x))^2}\$

\n

for a number $c$. You have to find $c$.

\n

$c=$ [[0]]

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Use the quotient rule to differentiate various functions.

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