// Numbas version: exam_results_page_options {"name": "Quotient Rule (Instructional)", "metadata": {"description": "Designed to instill a systematic method. The first 6 questions are scaffolded (step by step) followed by 2 randomly selected questions that only ask for a final answer.", "licence": "All rights reserved"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Scaffolded Questions", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Quotient Rule 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=0$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriate substitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (0) - (\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra skills to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*0-({aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

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Part a) Constant term for numerator

", "templateType": "randrange", "can_override": false}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

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Part a) x power for denominator

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Differentiate  $ \\large y=\\frac{\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dx}=   $[[2]]                    $  \\large\\frac{dv}{dx}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dx}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

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0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}x^{{aP}}-{aC2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "0", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aCF}x^{{aP}}-{aC2})^{2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], 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"typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aP2}*{aCF2}x^({aP2}-1) }$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriate substitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (\\simplify{{aP2}*{aCF2}x^({aP2}-1) }) - (\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra skills to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

\n
\n
\n
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Part a) Constant term for numerator

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Part a) x coefficient for denominator

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Part a) x power for denominator

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x coefficient for numerator

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Differentiate  $ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dx}=   $[[2]]                    $  \\large\\frac{dv}{dx}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dx}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

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"{aP2}*{aCF2}x^({aP2}-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}x^{{aP}}-{aC2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP2}*{aCF2}x^({aP2}-1)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF2}x^{aP2}+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aCF}x^{{aP}}-{aC2})^{2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quotient Rule 03", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_8w4NGYk.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_8w4NGYk.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF}e^(x)+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^(x)}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\simplify{{aCF}e^(x)+{aC1}}$                    $v=\\simplify{{aC2}-{aCF2}e^(x)}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Exponential Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives that gives  you:    $e^{kx}$    differentiates to     $k e^{kx}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aCF}e^(x) }$          and          $\\frac{dv}{dx}=\\simplify{ -{aCF2}e^(x) }$

\n

\n

 

\n

Make the appropriate substitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(\\var{aC2}-\\var{aCF2}e^(x)) \\times (\\var{aCF}e^(x)) - (\\var{aCF}e^(x)+\\var{aC1}) \\times  (-\\var{aCF2}e^(x)) }{(\\var{aC2}-\\var{aCF2}e^(x))^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra skills to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  ( {aC2}*{aCF}e^(x)+ {aC1}*{aCF2}e^(x))/(({aC2}-{aCF2}e^(x))^2)    }$ 

\n

\n

 

", "rulesets": {"std": ["all"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange", "can_override": false}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything", "can_override": false}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything", "can_override": false}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything", "can_override": false}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything", "can_override": false}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dx}=   $[[2]]                    $  \\large\\frac{dv}{dx}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dx}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)+{aC1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aC2}-{aCF2}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{aCF2}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aC2}-{aCF2}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{aCF2}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aC2}-{aCF2}e^(x))^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({aC2}*{aCF}e^(x)-{aCF2}e^(x)*{aCF}e^(x))-(({aCF}e^(x)*(-{aCF2}e^(x))+{aC1}*(-{aCF2}e^(x)))))/(({aC2}-{aCF2}e^(x))^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quotient Rule 04", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF}e^(x)+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^(x)}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\simplify{{aCF}e^(x)+{aC1}}$                    $v=\\simplify{{aC2}-{aCF2}e^(x)}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Exponential Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives that gives  you:    $e^{kx}$    differentiates to     $k e^{kx}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aCF}e^(x) }$          and          $\\frac{dv}{dx}=\\simplify{ -{aCF2}e^(x) }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(\\var{aC2}-\\var{aCF2}e^(x)) \\times (\\var{aCF}e^(x)) - (\\var{aCF}e^(x)+\\var{aC1}) \\times  (-\\var{aCF2}e^(x)) }{(\\var{aC2}-\\var{aCF2}e^(x))^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  ( {aC2}*{aCF}e^(x)+ {aC1}*{aCF2}e^(x))/(({aC2}-{aCF2}e^(x))^2)    }$ 

\n

\n

 

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Part a) Constant term for numerator

", "templateType": "randrange", "can_override": false}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

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Part a) x power for denominator

", "templateType": "anything", "can_override": false}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything", "can_override": false}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything", "can_override": false}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dx}=   $[[2]]                    $  \\large\\frac{dv}{dx}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dx}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)+{aC1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aC2}-{aCF2}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{aCF2}e^(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aC2}-{aCF2}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}e^(x)+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{aCF2}e^(x)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aC2}-{aCF2}e^(x))^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({aC2}*{aCF}e^(x)-{aCF2}e^(x)*{aCF}e^(x))-(({aCF}e^(x)*(-{aCF2}e^(x))+{aC1}*(-{aCF2}e^(x)))))/(({aC2}-{aCF2}e^(x))^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quotient Rule 05", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_W9at6dx.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_W9at6dx.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}} \\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=e^{\\var{aP}t}$                    $v=t+\\var{aC1}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Exponential Rule and for $v$ we need the Power Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives.

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{{aP}e^({aP}t)}$          and          $\\frac{dv}{dx}=1$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(t+\\var{aC1}) \\times (\\simplify{{aP}e^({aP}t)}) - e^{\\var{aP}t} }{(t+\\var{aC1})^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\frac{\\simplify{{aP}e^({aP}t)}(t+\\var{aC1}) - (e^{\\var{aP}t}) }{(t+\\var{aC1})^2}$ 

\n

\n

 

", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for denominator

\n
\n
\n
", "templateType": "randrange"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aP"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dt}=   $[[2]]                    $  \\large\\frac{dv}{dt}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dt}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "e^({aP}t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "t+{aC1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}e^({aP}t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "t+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}e^({aP}t)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "e^({aP}t)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "1", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(t+{aC1})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({aP}e^({aP}t)*(t+{aC1} )-e^({aP}t))/(t+{aC1})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": false}]}, {"name": "Quotient Rule 06", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aP2}*{aCF2}x^({aP2}-1) }$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (\\simplify{{aP2}*{aCF2}x^({aP2}-1) }) - (\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

\n
\n
\n
", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything"}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

First identify the two functions  $u$  and  $v$:

\n

$u=$[[0]]                    $v=$[[1]]

\n

 

\n

Now differentiate each one:

\n

$  \\large \\frac{du}{dx}=   $[[2]]                    $  \\large\\frac{dv}{dx}=   $[[3]]

\n

 

\n

Then using:

\n

$  \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $

\n

\n

Substitute each component into the formula in the correct place:

\n\n\n\n\n\n\n\n\n
                                                               $  \\Large \\frac{dy}{dx}=$[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]][[8]]
\n

\n

\n

\n

\n

\n

Finally tidy this up to give your final answer:

\n

$  \\Large \\frac{dy}{dx}=   $ [[9]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aCF2}x^{aP2}+{aC1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aCF}x^{{aP}}-{aC2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP2}*{aCF2}x^({aP2}-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aCF}x^{{aP}}-{aC2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP2}*{aCF2}x^({aP2}-1)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aCF2}x^{aP2}+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}*{aCF}x^{{aP}-1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({aCF}x^{{aP}}-{aC2})^{2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}]}]}, {"name": "Non-Scaffolded Questions", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Quotient Rule 01 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=0$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (0) - (\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*0-({aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

\n

$  \\Large \\frac{dy}{dx}=   $[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(({aCF}x^{{aP}}-{aC2})*0-({aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}]}, {"name": "Quotient Rule 02 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aP2}*{aCF2}x^({aP2}-1) }$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (\\simplify{{aP2}*{aCF2}x^({aP2}-1) }) - (\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

\n
\n
\n
", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything"}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

\n

$  \\Large \\frac{dy}{dx}=   $[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}]}, {"name": "Quotient Rule 03 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_8w4NGYk.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_8w4NGYk.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\simplify{{aCF}e^{x}+{aC1}}$                    $v=\\simplify{{aC2}-{aCF2}e^{x}}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Exponential Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives that gives  you:    $e^{kx}$    differentiates to     $k e^{kx}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aCF}e^{x} }$          and          $\\frac{dv}{dx}=\\simplify{ -{aCF2}e^{x} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(\\var{aC2}-\\var{aCF2}e^{x}) \\times (\\var{aCF}e^{x}) - (\\var{aCF}e^{x}+\\var{aC1}) \\times  (-\\var{aCF2}e^{x}) }{(\\var{aC2}-\\var{aCF2}e^{x})^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  ( {aC2}*{aCF}e^{x}+ {aC1}*{aCF2}e^{x})/(({aC2}-{aCF2}e^{x})^2)    }$ 

\n

\n

 

", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything"}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}$

\n

\n

\n

$  \\Large \\frac{dy}{dx}=   $[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(({aC2}*{aCF}e^{x}-{aCF2}e^{x}*{aCF}e^{x})-(({aCF}e^{x}*(-{aCF2}e^{x})+{aC1}*(-{aCF2}e^{x}))))/(({aC2}-{aCF2}e^{x})^2)", "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}]}, {"name": "Quotient Rule 04 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\simplify{{aCF}e^{x}+{aC1}}$                    $v=\\simplify{{aC2}-{aCF2}e^{x}}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Exponential Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives that gives  you:    $e^{kx}$    differentiates to     $k e^{kx}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aCF}e^{x} }$          and          $\\frac{dv}{dx}=\\simplify{ -{aCF2}e^{x} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(\\var{aC2}-\\var{aCF2}e^{x}) \\times (\\var{aCF}e^{x}) - (\\var{aCF}e^{x}+\\var{aC1}) \\times  (-\\var{aCF2}e^{x}) }{(\\var{aC2}-\\var{aCF2}e^{x})^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  ( {aC2}*{aCF}e^{x}+ {aC1}*{aCF2}e^{x})/(({aC2}-{aCF2}e^{x})^2)    }$ 

\n

\n

 

", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

", "templateType": "anything"}, "aP2": {"name": "aP2", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1 except aP)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aCF", "aP", "aC2", "aCF2", "aP2"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{\\var{aCF}e^{x}+\\var{aC1}}{\\var{aC2}-\\var{aCF2}e^{x}}$

\n

\n

\n

$  \\Large \\frac{dy}{dx}=   $[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(({aC2}*{aCF}e^{x}-{aCF2}e^{x}*{aCF}e^{x})-(({aCF}e^{x}*(-{aCF2}e^{x})+{aC1}*(-{aCF2}e^{x}))))/(({aC2}-{aCF2}e^{x})^2)", "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}]}, {"name": "Quotient Rule 05 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_W9at6dx.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_W9at6dx.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}} \\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=e^{\\var{aP}t}$                    $v=t+\\var{aC1}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Exponential Rule and for $v$ we need the Power Rule:

\n

\n

if  $  \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$

\n

\n

If you don't follow this, use your Table of Derivatives.

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{{aP}e^({aP}t)}$          and          $\\frac{dv}{dx}=1$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}=  \\frac{(t+\\var{aC1}) \\times (\\simplify{{aP}e^({aP}t)}) - e^{\\var{aP}t} }{(t+\\var{aC1})^2}  $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

\n

$ \\large \\frac{dy}{dx}= \\frac{\\simplify{{aP}e^({aP}t)}(t+\\var{aC1}) - (e^{\\var{aP}t}) }{(t+\\var{aC1})^2}$ 

\n

\n

 

", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for denominator

\n
\n
\n
", "templateType": "randrange"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aP"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Differentiate  $ \\large y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}}$

\n

\n

\n

$  \\Large \\frac{dy}{dx}=   $[[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({aP}e^({aP}t)*(t+{aC1} )-e^({aP}t))/(t+{aC1})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": false}]}, {"name": "Quotient Rule 06 (non-scaffolded)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "

Instructional \"drill\" exercise to emphasize the method.

Thanks to Christian for his method for use of gaps in fractions.

", "licence": "All rights reserved"}, "statement": "

We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:

\n

\n

If  $\\large y=\\frac{u}{v}$  then:

\n

\\[   \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]

\n

\n
\n

Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.

", "advice": "

We are asked to differentiate:

\n

\\[ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}\\]

\n

\n

Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.

\n

\n

$u$ is the numerator, the function \"on top\", $v$ is the denominator, the function \"on the bottom\".

\n

\n

$u=\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}$                    $v=\\var{aCF}x^{\\var{aP}}-\\var{aC2}$

\n

 

\n

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule:

\n

\n

if  $y=a x^{n}$          then          $\\frac{dy}{dx}=n \\times a x^{n-1}$

\n

\n

Applying this gives us:

\n

$\\large \\frac{du}{dx}=\\simplify{ {aP2}*{aCF2}x^({aP2}-1) }$          and          $\\frac{dv}{dx}=\\simplify{ {aP}  {aCF}x^{{aP}-1} }$

\n

\n

 

\n

Make the appropriatesubstitutions into the formula:

\n

\n

$ \\large  \\frac{dy}{dx}= \\frac{({\\var{aCF}x^{\\var{aP}}-\\var{aC2}}) \\times (\\simplify{{aP2}*{aCF2}x^({aP2}-1) }) - (\\var{aCF2}x^\\var{{aP2}}+\\var{aC1}) \\times (\\simplify{{aP}  {aCF}x^{{aP}-1} })}{(\\var{aCF}x^{\\var{aP}}-\\var{aC2})^{2}} $

\n

\n

 

\n

Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:

\n

\n

$ \\large \\frac{dy}{dx}= \\simplify{  (({aCF}x^{{aP}}-{aC2})*({aP2}*{aCF2}x^({aP2}-1))-({aCF2}x^{aP2}+{aC1})*({aP}*{aCF}x^{{aP}-1}))/(({aCF}x^{{aP}}-{aC2})^{2}) }$

\n

 

\n

 

\n
\n
\n
", "rulesets": {"std": ["all"]}, "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "

Part a) Constant term for numerator

", "templateType": "randrange"}, "aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

Part a) x coefficient for denominator

", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aCF except aC1)", "description": "

Part a) x power for denominator

", "templateType": "anything"}, "aC2": {"name": "aC2", "group": "Part (a)", "definition": "random(2..9 except aCF)", "description": "", "templateType": "anything"}, "aCF2": {"name": "aCF2", "group": "Part (a)", "definition": "random(2..6 except aC1)", "description": "

x coefficient for numerator

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Differentiate  $ \\large y=\\frac{\\var{aCF2}x^{\\var{aP2}}+\\var{aC1}}{\\var{aCF}x^{\\var{aP}}-\\var{aC2}}$

\n

\n

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$  \\Large \\frac{dy}{dx}=   $[[0]]

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