// Numbas version: exam_results_page_options {"name": "Product 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": "Product Rule 01", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_TcDOo33.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_TcDOo33.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 PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ \\large y=\\var{aCF}x^{\\var{aP}} \\sin{(x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=\\var{aCF}x^{\\var{aP}}$ $v=\\sin{(x)}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Power Rule and $v$ can be done using your Table of Derivatives:

Applying these methods gives us:

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

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF}x^{\\var{aP}} \\times \\cos{(x)} + \\sin{(x)} \\times \\simplify {{aP}*{aCF}x^{{aP}-1} } $

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

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

", "rulesets": {}, "variables": {"aCF": {"name": "aCF", "group": "part (a)", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange"}, "aP": {"name": "aP", "group": "part (a)", "definition": "random(2..4 except aCF)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "part (a)", "variables": ["aP", "aCF"]}], "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=\\var{aCF}x^{\\var{aP}} \\sin{(x)}$

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

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

Now differentiate each one:

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

Then using:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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": "\u00a0{aCF}x^{{aP}}", "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", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "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": ""}]}, {"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, "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": "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "\u00a0{aCF}x^{{aP}}", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos(x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sin(x)", "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": "0.5", "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, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aCF}x^{{aP}})cos(x)+sin(x)({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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Product Rule 02", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_0EqjitN.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_0EqjitN.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 PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ \\large y=x^{\\var{b}} \\ln{(\\var{b}x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=x^{\\var{b}}$ $v=\\ln{(\\var{b}x)}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Power Rule and $v$ can be done using your Table of Derivatives:

Applying these methods gives us:

$\\large \\frac{du}{dx}=\\simplify{ {b} * x^{{{b}}-1} } $ and $ \\large \\frac{dv}{dx}=\\frac{1}{x}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= x^{\\var{b}} \\times \\frac{1}{x} + \\ln{(\\var{b}x)} \\times \\simplify{{b} *x^{{{b}}-1} } $

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

$ \\large \\frac{dy}{dx}= \\simplify{x^{{b}} * x^(-1)} + \\simplify{{b} *x^{{{b}}-1} } \\ln{(\\var{b}x)}$

only variable for part b - used twice

", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b"], "variable_groups": [{"name": "part (a)", "variables": ["aP", "aCF"]}], "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=x^{\\var{b}} \\ln{(\\var{b}x)}$

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

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

Now differentiate each one:

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

Then using:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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": "x^{b} ", "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", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln({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": "{b}x^({b}-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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^{b} ", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/x", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln({b}x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}x^({b}-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": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x^{b})(1/x)+ln({b}x)*{b}x^({b}-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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Product Rule 03", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives.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 PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ y=e^{\\var{a}x} \\ln{(\\var{b}x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=e^{\\var{a}x} $ $v=\\ln{(\\var{b}x)}$

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

This gives us:

$\\large \\frac{du}{dx}=\\var{a}e^{\\var{a}x} $ and $ \\large \\frac{dv}{dx}= \\frac{1}{x}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= e^{\\var{a}x} \\times \\frac{1}{x} + \\ln{(\\var{b}x)} \\times \\var{a}e^{\\var{a}x} $

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

$ \\large \\frac{dy}{dx}= \\frac{ e^{\\var{a}x}}{x} + \\var{a}e^{\\var{a}x} \\ln{(\\var{b}x)} $

", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9 except a)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "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=e^{\\var{a}x} \\ln{(\\var{b}x)}$

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

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

Now differentiate each one:

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

Then using:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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": "e^({a}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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln({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": "{a}*e^({a}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/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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^({a}x) ", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1/x", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln({b}x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}*e^({a}x) ", "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": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(e^({a}x))*(1/x)+(ln({b}x))*({a}*e^({a}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": "Product Rule 04", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_p6SUaKT.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_p6SUaKT.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 questions (non-randomized) to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\frac{e^{\\var{b}x}}{x^{\\var{a}}} \\]

At first this function looks like two functions divided rather than multiplied, making it a candidate for the Quotient Rule instead of the Product Rule.

However, recognising that the function $\\frac{1}{x^{\\var{a}}}$ (the denominator) can be rewritten as $x^{-\\var{a}}$ gives us:

\\[ y=e^{\\var{b}x} x^{-\\var{a}}\\]

We are now on much more familiar ground for the Product Rule.

$u$ is the first function, $v$ is second:

$\\large u=e^{\\var{b}x} $ $ \\large v=x^{-\\var{a}}$

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

This gives us:

$\\large \\frac{du}{dx}=\\var{b}e^{\\var{b}x} $ and $ \\large \\frac{dv}{dx}= \\simplify{ -{a}x^{-{a}-1} }$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= e^{\\var{b}x} \\times \\simplify{-{a}x^{-{a}-1} } + x^{-\\var{a}} \\times \\var{b}e^{\\var{b}x} $

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

$ \\large \\frac{dy}{dx}= -\\frac{\\var{a} e^{\\var{b}x} }{\\simplify{x^{{a}+1}}}+\\frac{\\var{b}e^{\\var{b}x}}{x^{\\var{a}}}$

Notice that, whenever possible, your final answer should not contain negative indices (powers).

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "

", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6 except b)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "a"], "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=\\frac{e^{\\var{b}x}}{x^{\\var{a}}}$

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

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

Now differentiate each one:

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

Then using:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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": "e^({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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^{-{a}}", "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": "{b}e^({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": "-{a}x^(-{a}-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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^({b}x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{a}x^(-{a}-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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^{-{a}}", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}e^({b}x)", "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": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(e^({b}x))*(-{a}x^(-{a}-1))+(x^{-{a}})*({b}e^({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": "Product Rule 05", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_BJvHVn2.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_BJvHVn2.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 questions (non-randomized) to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\var{aCF}x^{\\var{aP}} e^{\\var{eP}x} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$\\large u=\\var{aCF}x^{\\var{aP}} $ $\\large v=e^{\\var{eP}x} $

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

This gives us:

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

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF}x^{\\var{aP}} \\times \\simplify{{eP}*e^({eP} x)} + e^{\\var{eP}x} \\times \\simplify{{aP}*{aCF}*x^({aP}-1) } $

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

$ \\large \\frac{dy}{dx}= \\simplify{{aCF}x^{{aP}} * {eP}*e^({eP} x)} + \\simplify{{aP}*{aCF}*x^{{aP}-1} } e^{\\var{eP}x} $

", "rulesets": {}, "variables": {"aCF": {"name": "aCF", "group": "Part (a)", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..6 except aCF)", "description": "", "templateType": "anything"}, "eP": {"name": "eP", "group": "Part (a)", "definition": "random(2..6 except aP)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aCF", "aP", "eP"]}], "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=\\var{aCF}x^{\\var{aP}} e^{\\var{eP}x} $

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

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

Now differentiate each one:

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

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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}x^{{aP}}", "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", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^({eP} 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": "{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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{eP}*e^({eP} 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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aCF}x^{{aP}}", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{eP}*e^({eP} x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^({eP} x)", "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": "0.5", "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, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({aCF}x^{{aP}})*({eP}*e^({eP} x))+(e^({eP} x))*({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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Product Rule 06", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_UV2rNbD.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_UV2rNbD.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 questions (non-randomized) to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

", "advice": "

We are asked to differentiate:

\\[ y=\\var{bCF}x^{\\var{bP1}} \\cos{(\\var{bCF2}x)}\\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$\\large u=\\var{bCF}x^{\\var{bP1}} $ $\\large v=\\cos{(\\var{bCF2}x)} $

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

This gives us:

$\\large \\frac{du}{dx}=\\simplify{ {bP1}*{bCF}*x^({bP1}-1) }$ and $ \\large \\frac{dv}{dx}=- \\var{bCF2} \\sin{(\\var{bCF2}x)}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{bCF}x^{\\var{bP1}} \\times - \\var{bCF2}\\sin{(\\var{bCF2}x)} + \\cos{(\\var{bCF2}x)} \\times \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } $

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

$ \\large \\frac{dy}{dx}= \\simplify{ - {bCF2}* {bCF} x^{{bP1}}} \\sin{(\\var{bCF2}x)} + \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } \\cos{(\\var{bCF2}x)} $

", "rulesets": {}, "variables": {"bCF": {"name": "bCF", "group": "Part (b)", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "bP1": {"name": "bP1", "group": "Part (b)", "definition": "random(2..6 except bCF)", "description": "", "templateType": "anything"}, "bCF2": {"name": "bCF2", "group": "Part (b)", "definition": "random(2..6 except bCF)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (b)", "variables": ["bCF", "bP1", "bCF2"]}], "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=\\var{bCF}x^{\\var{bP1}} \\cos{(\\var{bCF2}x)}$

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

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

Now differentiate each one:

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

Substitute each component into the formula in the correct place:

$ \\large \\frac{dy}{dx}=$[[4]]$ \\large \\times$[[5]]$ \\large + $[[6]]$ \\large \\times$[[7]]

Finally tidy this up to give your final answer:

$ \\large \\frac{dy}{dx}= $[[8]]

", "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": "{bCF}x^{{bP1}} ", "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", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos({bCF2}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": "{bP1}*{bCF}x^{{bP1}-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, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{bCF2}*-sin({bCF2}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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{bCF}x^{{bP1}}", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{bCF2}*-sin({bCF2}x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos({bCF2}x)", "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": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{bP1}*{bCF}x^{{bP1}-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": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({bCF}x^{{bP1}})*({bCF2}*-sin({bCF2}x))+(cos({bCF2}x))*({bP1}*{bCF}x^{{bP1}-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": ""}]}], "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": "Product Rule 01 (non scaffold)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_TcDOo33.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_TcDOo33.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 PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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:

\\[ \\large y=\\var{aCF}x^{\\var{aP}} \\sin{(x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=\\var{aCF}x^{\\var{aP}}$ $v=\\sin{(x)}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Power Rule and $v$ can be done using your Table of Derivatives:

Applying these methods gives us:

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

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF}x^{\\var{aP}} \\times \\cos{(x)} + \\sin{(x)} \\times \\simplify {{aP}*{aCF}x^{{aP}-1} } $

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

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

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

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

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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:

\\[ \\large y=x^{\\var{b}} \\ln{(\\var{b}x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=x^{\\var{b}}$ $v=\\ln{(\\var{b}x)}$

Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the Power Rule and $v$ can be done using your Table of Derivatives:

Applying these methods gives us:

$\\large \\frac{du}{dx}=\\simplify{ {b} * x^{{{b}}-1} } $ and $ \\large \\frac{dv}{dx}=\\frac{1}{x}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= x^{\\var{b}} \\times \\frac{1}{x} + \\ln{(\\var{b}x)} \\times \\simplify{{b} *x^{{{b}}-1} } $

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

$ \\large \\frac{dy}{dx}= \\simplify{x^{{b}} * x^(-1)} + \\simplify{{b} *x^{{{b}}-1} } \\ln{(\\var{b}x)}$

only variable for part b - used twice

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

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

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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=e^{\\var{a}x} \\ln{(\\var{b}x)} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$u=e^{\\var{a}x} $ $v=\\ln{(\\var{b}x)}$

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

This gives us:

$\\large \\frac{du}{dx}=\\var{a}e^{\\var{a}x} $ and $ \\large \\frac{dv}{dx}= \\frac{1}{x}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= e^{\\var{a}x} \\times \\frac{1}{x} + \\ln{(\\var{b}x)} \\times \\var{a}e^{\\var{a}x} $

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

$ \\large \\frac{dy}{dx}= \\frac{ e^{\\var{a}x}}{x} + \\var{a}e^{\\var{a}x} \\ln{(\\var{b}x)} $

Differentiate $ y=e^{\\var{a}x} \\ln{(\\var{b}x)}$

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

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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=\\frac{e^{\\var{b}x}}{x^{\\var{a}}} \\]

At first this function looks like two functions divided rather than multiplied, making it a candidate for the Quotient Rule instead of the Product Rule.

However, recognising that the function $\\frac{1}{x^{\\var{a}}}$ (the denominator) can be rewritten as $x^{-\\var{a}}$ gives us:

\\[ y=e^{\\var{b}x} x^{-\\var{a}}\\]

We are now on much more familiar ground for the Product Rule.

$u$ is the first function, $v$ is second:

$\\large u=e^{\\var{b}x} $ $ \\large v=x^{-\\var{a}}$

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

This gives us:

$\\large \\frac{du}{dx}=\\var{b}e^{\\var{b}x} $ and $ \\large \\frac{dv}{dx}= \\simplify{ -{a}x^{-{a}-1} }$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= e^{\\var{b}x} \\times \\simplify{-{a}x^{-{a}-1} } + x^{-\\var{a}} \\times \\var{b}e^{\\var{b}x} $

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

$ \\large \\frac{dy}{dx}= -\\frac{\\var{a} e^{\\var{b}x} }{\\simplify{x^{{a}+1}}}+\\frac{\\var{b}e^{\\var{b}x}}{x^{\\var{a}}}$

Notice that, whenever possible, your final answer should not contain negative indices (powers).

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "

Differentiate $ y=\\frac{e^{\\var{b}x}}{x^{\\var{a}}}$

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

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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{aP}} e^{\\var{eP}x} \\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$\\large u=\\var{aCF}x^{\\var{aP}} $ $\\large v=e^{\\var{eP}x} $

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

This gives us:

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

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{aCF}x^{\\var{aP}} \\times \\simplify{{eP}*e^({eP} x)} + e^{\\var{eP}x} \\times \\simplify{{aP}*{aCF}*x^({aP}-1) } $

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

$ \\large \\frac{dy}{dx}= \\simplify{{aCF}x^{{aP}} * {eP}*e^({eP} x)} + \\simplify{{aP}*{aCF}*x^{{aP}-1} } e^{\\var{eP}x} $

Differentiate $ y=\\var{aCF}x^{\\var{aP}} e^{\\var{eP}x} $

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

We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:

If $y=u \\times v$ then:

\\[ \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]

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{bCF}x^{\\var{bP1}} \\cos{(\\var{bCF2}x)}\\]

Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.

$u$ is the first function, $v$ is second:

$\\large u=\\var{bCF}x^{\\var{bP1}} $ $\\large v=\\cos{(\\var{bCF2}x)} $

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

This gives us:

$\\large \\frac{du}{dx}=\\simplify{ {bP1}*{bCF}*x^({bP1}-1) }$ and $ \\large \\frac{dv}{dx}=- \\var{bCF2} \\sin{(\\var{bCF2}x)}$

We now use the formula:

$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $

Make the appropriate substitutions into the formula:

$ \\large \\frac{dy}{dx}= \\var{bCF}x^{\\var{bP1}} \\times - \\var{bCF2}\\sin{(\\var{bCF2}x)} + \\cos{(\\var{bCF2}x)} \\times \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } $

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

$ \\large \\frac{dy}{dx}= \\simplify{ - {bCF2}* {bCF} x^{{bP1}}} \\sin{(\\var{bCF2}x)} + \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } \\cos{(\\var{bCF2}x)} $

Differentiate $ y=\\var{bCF}x^{\\var{bP1}} \\cos{(\\var{bCF2}x)}$

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

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "8", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({bCF}x^{{bP1}})*({bCF2}*-sin({bCF2}x))+(cos({bCF2}x))*({bP1}*{bCF}x^{{bP1}-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": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_TcDOo33.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_TcDOo33.pdf"], ["question-resources/Table_of_Derivatives_0EqjitN.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_0EqjitN.pdf"], ["question-resources/Table_of_Derivatives.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives.pdf"], ["question-resources/Table_of_Derivatives_p6SUaKT.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_p6SUaKT.pdf"], ["question-resources/Table_of_Derivatives_BJvHVn2.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_BJvHVn2.pdf"], ["question-resources/Table_of_Derivatives_UV2rNbD.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_UV2rNbD.pdf"]]}