// Numbas version: exam_results_page_options {"name": "Product Rule 05 (non scaffold)", "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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Product Rule 05 (non scaffold)", "tags": [], "metadata": {"description": "Instructional questions (non-randomized) to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "
Now it is time to get out your paper and pencil and try similar questions without the help. Carry out the same steps, lay it out the same way but you only need to input the final answer.
", "advice": "We are asked to differentiate:
\n\\[ y=\\var{aCF}x^{\\var{aP}} e^{\\var{eP}x} \\]
\n\nRecognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.
\n\n$u$ is the first function, $v$ is second:
\n\n$\\large u=\\var{aCF}x^{\\var{aP}} $ $\\large v=e^{\\var{eP}x} $
\n\n
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:
\n\nThis gives us:
\n$\\large \\frac{du}{dx}=\\simplify{ {aP}*{aCF}*x^({aP}-1) }$ and $ \\large \\frac{dv}{dx}= \\simplify{{eP}*e^({eP} x)}$
\n\nWe now use the formula:
\n$ \\large \\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $
\nMake the appropriate substitutions into the formula:
\n\n$ \\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) } $
\n\n\n
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:
\n\n$ \\large \\frac{dy}{dx}= \\simplify{{aCF}x^{{aP}} * {eP}*e^({eP} x)} + \\simplify{{aP}*{aCF}*x^{{aP}-1} } e^{\\var{eP}x} $
\n", "rulesets": {}, "extensions": [], "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": "
$ \\large \\frac{dy}{dx}= $[[0]]
\n", "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": "({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", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}