// Numbas version: exam_results_page_options {"name": "Chain 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": "Creative Commons Attribution-NonCommercial 4.0 International"}, "duration": 0, "percentPass": "70", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Scaffolded Questions", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Chain 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.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=(\\var{xCF}x^{\\var{xP}}-\\var{C})^{\\var{P}} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{xCF}x^{\\var{xP}}-\\var{C}$ then $y=u^{\\var{P}}$

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

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{xCF2}x^{{xP2}}}$ and $ \\large \\frac{dy}{du}= \\simplify{{P}u^{{P2}}}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\simplify{{xCF2}x^{{xP2}}} \\times \\simplify{{P}u^{{P2}}} $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\simplify{ ({xCF2}x^{xP2})*({P}u^{P2}) }$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\simplify{({xCF2}x^{xP2})*({P}({xCF}x^{xP}-{C})^{P2})}$

", "rulesets": {}, "variables": {"xCF": {"name": "xCF", "group": "Part (a)", "definition": "random(2 .. 5#1)", "description": "

Part a) x co-efficient

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Part a) x power

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Part a) constant

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Part a) power the bracket is raised to

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differentiated x power

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differentiated bracket power

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Part b) x power

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Differentiate $ y=(\\var{xCF}x^{\\var{xP}}-\\var{C})^{\\var{P}} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

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"nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{P}u^{P2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "({xCF2}x^{xP2})*({P}u^{P2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, 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We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\cos{(x^{\\var{bP}})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=x^{\\var{bP}}$ then $y=cos(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first we need the Power Rule, and for the second you can use your Table of Derivatives.

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{bP}x^{bP2}}$ and $ \\large \\frac{dy}{du}= -sin(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\simplify{{bP}x^{bP2}} \\times -sin(u)$

Which simplifies to:

$ \\large \\frac{dy}{dx}=-\\simplify{ {bP}x^{bP2}}sin(u) $

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=-\\simplify{{bP}x^{bP2}} sin(x^{\\var{bP}})$

", "rulesets": {}, "variables": {"bP": {"name": "bP", "group": "Part (b)", "definition": "random(2 .. 5#1)", "description": "

Part b) x power

", "templateType": "randrange"}, "bP2": {"name": "bP2", "group": "Part (b)", "definition": "bP-1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (b)", "variables": ["bP", "bP2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "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 $ y=\\cos{(x^{\\var{bP}})} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

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"nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-sin(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "({bP}x^{bP2})*-sin(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 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"partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Chain Rule 03", "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.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=(\\var{aCF}x-\\var{C})^{\\var{aP}} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{aCF}x-\\var{C}$ then $y=u^{\\var{aP}}$

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

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{aCF}$ and $ \\large \\frac{dy}{du}= \\simplify{{aP}u^({aP}-1)}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF} \\times \\simplify{{aP}u^({aP}-1)} $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\simplify{ {aCF}*{aP}u^({aP}-1) }$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\simplify{ {aCF}*{aP}({aCF}x-{C})^({aP}-1) }$

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Part a) x coefficient

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Differentiate $ y=(\\var{aCF}x-\\var{C})^{\\var{aP}} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

", "gaps": [{"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}x-{C}", "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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], 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"marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aP}u^({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": "u", "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}*{aP}u^({aP}-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, 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[], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Chain Rule 04", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_3aXLjOU.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_3aXLjOU.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.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\sin{(\\var{b}x)} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{b}x$ then $y=sin(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first of these we need the Power Rule and for the second, your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{b}$ and $ \\large \\frac{dy}{du}= cos(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{b}\\times cos(u) $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\var{b} cos(u)$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\var{b}cos(\\var{b}x)$

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Part (b)", "definition": "random(-9..9 except -1 except 0 except 1 except -2 except 2)", "description": "

Part b) x coefficient

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (b)", "variables": ["b"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "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 $ y=\\sin{(\\var{b}x)} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

", "gaps": [{"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": "{b}x", "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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sin(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{b}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{b}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{b}*cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{b}*cos({b}x)", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Chain Rule 05", "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.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\sin{(\\simplify{{CF1}x}+e^{x})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\simplify{{CF1}x}+e^{x}$ then $y=sin(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first of these we need the Power Rule and for the second, your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{CF1}}+e^{x}$ and $ \\large \\frac{dy}{du}= cos(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= (\\simplify{{CF1}}+e^{x}) \\times cos(u) $

Which simplifies to:

$ \\large \\frac{dy}{dx}=(\\simplify{{CF1}}+e^{x}) cos(u)$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=(\\simplify{{CF1}}+e^{x}) cos(\\simplify{{CF1}x}+e^{x})$

", "rulesets": {}, "variables": {"CF1": {"name": "CF1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["CF1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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 $ y=\\sin{(\\simplify{{CF1}x}+e^{x})} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

", "gaps": [{"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": "{CF1}x+e^(x)", "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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sin(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{CF1}+e^{x}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "{CF1}+e^{x}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "({CF1}+e^{x})*cos(u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, 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[], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Chain Rule 06", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_zWorKza.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_zWorKza.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.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\ln({\\simplify{{CF}x}+\\sin{(x)})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\simplify{{CF}x+sin(x)}$ then $y=\\ln(u)$

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule and your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{CF}+cos(x)$ and $ \\large \\frac{dy}{du}= \\frac{1}{u}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= (\\var{CF}+cos(x)) \\times \\frac{1}{u}$

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{u}$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{\\simplify{{CF}x+sin(x)}}$

", "rulesets": {}, "variables": {"CF": {"name": "CF", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["CF"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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 $ y=\\ln({\\var{CF}x+\\sin{(x)})} $

First identify the \"innermost\" function, and substitute $u$:

Let $u=$[[0]] Then $y=$[[1]]

Then:

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

Now using:

$\\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du}$

$\\large \\frac{dy}{dx}=$[[4]]$\\times$[[5]]

Which simplifies to:

$\\large \\frac{dy}{dx}=$[[6]]

Remember that $u$ was a variable that we introduced and not part of the original problem.

Replace $u$ from our substitution to give the final answer:

$\\large \\frac{dy}{dx}=$[[7]]

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"customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{CF}+cos(x)", "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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/u", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "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": "({CF}+cos(x))*(1/u)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "u", "value": ""}, {"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": "({CF}+cos(x))/({CF}x+sin(x))", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Non-Scaffolded Questions", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Chain 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.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=(\\var{xCF}x^{\\var{xP}}-\\var{C})^{\\var{P}} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{xCF}x^{\\var{xP}}-\\var{C}$ then $y=u^{\\var{P}}$

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

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{xCF2}x^{{xP2}}}$ and $ \\large \\frac{dy}{du}= \\simplify{{P}u^{{P2}}}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\simplify{{xCF2}x^{{xP2}}} \\times \\simplify{{P}u^{{P2}}} $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\simplify{ ({xCF2}x^{xP2})*({P}u^{P2}) }$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\simplify{({xCF2}x^{xP2})*({P}({xCF}x^{xP}-{C})^{P2})}$

", "rulesets": {}, "variables": {"xCF": {"name": "xCF", "group": "Part (a)", "definition": "random(2 .. 5#1)", "description": "

Part a) x co-efficient

", "templateType": "randrange"}, "xP": {"name": "xP", "group": "Part (a)", "definition": "random(2 .. 5#1)", "description": "

Part a) x power

", "templateType": "randrange"}, "C": {"name": "C", "group": "Part (a)", "definition": "random(1 .. 9#1)", "description": "

Part a) constant

", "templateType": "randrange"}, "P": {"name": "P", "group": "Part (a)", "definition": "random(2 .. 5#1)", "description": "

Part a) power the bracket is raised to

differentiated x power

", "templateType": "anything"}, "P2": {"name": "P2", "group": "Part (a)", "definition": "P-1", "description": "

differentiated bracket power

", "templateType": "anything"}, "bP": {"name": "bP", "group": "Part (b)", "definition": "random(2 .. 5#1)", "description": "

Part b) x power

Differentiate $ y=(\\var{xCF}x^{\\var{xP}}-\\var{C})^{\\var{P}} $

$\\large \\frac{dy}{dx}=$[[0]]

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=\\cos{(x^{\\var{bP}})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=x^{\\var{bP}}$ then $y=cos(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first we need the Power Rule, and for the second you can use your Table of Derivatives.

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{bP}x^{bP2}}$ and $ \\large \\frac{dy}{du}= -sin(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\simplify{{bP}x^{bP2}} \\times -sin(u)$

Which simplifies to:

$ \\large \\frac{dy}{dx}=-\\simplify{ {bP}x^{bP2}}sin(u) $

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=-\\simplify{{bP}x^{bP2}} sin(x^{\\var{bP}})$

", "rulesets": {}, "variables": {"bP": {"name": "bP", "group": "Part (b)", "definition": "random(2 .. 5#1)", "description": "

Part b) x power

", "templateType": "randrange"}, "bP2": {"name": "bP2", "group": "Part (b)", "definition": "bP-1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (b)", "variables": ["bP", "bP2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "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 $ y=\\cos{(x^{\\var{bP}})} $

$\\large \\frac{dy}{dx}=$[[0]]

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=(\\var{aCF}x-\\var{C})^{\\var{aP}} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{aCF}x-\\var{C}$ then $y=u^{\\var{aP}}$

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

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{aCF}$ and $ \\large \\frac{dy}{du}= \\simplify{{aP}u^({aP}-1)}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF} \\times \\simplify{{aP}u^({aP}-1)} $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\simplify{ {aCF}*{aP}u^({aP}-1) }$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\simplify{ {aCF}*{aP}({aCF}x-{C})^({aP}-1) }$

", "rulesets": {}, "variables": {"aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(5 .. 10#1)", "description": "

Part a) x coefficient

Differentiate $ y=(\\var{aCF}x-\\var{C})^{\\var{aP}} $

$\\large \\frac{dy}{dx}=$[[0]]

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=\\sin{(\\var{b}x)} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\var{b}x$ then $y=sin(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first of these we need the Power Rule and for the second, your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{b}$ and $ \\large \\frac{dy}{du}= cos(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{b}\\times cos(u) $

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\var{b} cos(u)$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\var{b}cos(\\var{b}x)$

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Part (b)", "definition": "random(-9..9 except -1 except 0 except 1 except -2 except 2)", "description": "

Part b) x coefficient

Differentiate $ y=\\sin{(\\var{b}x)} $

$\\large \\frac{dy}{dx}=$[[0]]

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=\\sin{(\\simplify{{CF1}x}+e^{x})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\simplify{{CF1}x}+e^{x}$ then $y=sin(u)$

Now, we need to use the approriate techniques to differentiate each of these, for the first of these we need the Power Rule and for the second, your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\simplify{{CF1}}+e^{x}$ and $ \\large \\frac{dy}{du}= cos(u)$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= (\\simplify{{CF1}}+e^{x}) \\times cos(u) $

Which simplifies to:

$ \\large \\frac{dy}{dx}=(\\simplify{{CF1}}+e^{x}) cos(u)$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=(\\simplify{{CF1}}+e^{x}) cos(\\simplify{{CF1}x}+e^{x})$

Differentiate $ y=\\sin{(\\simplify{{CF1}x}+e^{x})} $

$\\large \\frac{dy}{dx}=$[[0]]

We use the CHAIN RULE (also called the FUNCTION OF A FUNCTION RULE) when the function that we need to differentiate is actually one function \"nested\" inside another.

If $y=f(g(x))$ to find $\\frac{dy}{dx}$ , we need to do two things::

Substitute $u=g(x)$ so that $y=f(u)$
Use the Chain Rule:

\\[ \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} \\]

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:

\\[ y=\\ln({\\simplify{{CF}x}+\\sin{(x)})} \\]

Recognising that this is a \"function of another function\", we need to identify the \"innermost\" of the two functions that are involved and substitute $u$

Let $u=\\simplify{{CF}x+sin(x)}$ then $y=\\ln(u)$

Now, we need to use the approriate techniques to differentiate each of these, for these functions we need the Power Rule and your Table of Derivatives.:

Applying this method gives us:

$\\large \\frac{du}{dx}=\\var{CF}+cos(x)$ and $ \\large \\frac{dy}{du}= \\frac{1}{u}$

We now use the Chain Rule formula:

$ \\large \\large \\frac{dy}{dx}=\\frac{du}{dx} \\times \\frac{dy}{du} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= (\\var{CF}+cos(x)) \\times \\frac{1}{u}$

Which simplifies to:

$ \\large \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{u}$

Now, finally, we must remember that $u$ was a variable that we introduced and was not part of the original problem.

Replace $u$ from our original substitution to give the final answer:

$\\large \\frac{dy}{dx}=\\frac{\\var{CF}+cos(x)}{\\simplify{{CF}x+sin(x)}}$

Differentiate $ y=\\ln({\\var{CF}x+\\sin{(x)})} $

$\\large \\frac{dy}{dx}=$[[0]]

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