// Numbas version: finer_feedback_settings
{"name": "Product Rule 03 (non scaffold)", "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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Product Rule 03 (non scaffold)", "tags": [], "metadata": {"description": "Instructional \"drill\" exercise to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4>We use the PRODUCT RULE when the function that we need to differentiate is actually two functions multiplied together:</h4>\n<h4></h4>\n<h4>If&nbsp; $y=u \\times v$&nbsp; then:</h4>\n<h4>\\[&nbsp; &nbsp;\\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} \\]</h4>\n<hr/>\n<p>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.</p>", "advice": "<p>We are asked to differentiate:</p>\n<p>\\[&nbsp;&nbsp;y=e^{\\var{a}x} \\ln{(\\var{b}x)}&nbsp;\\]</p>\n<p></p>\n<p>Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.</p>\n<p></p>\n<p>$u$ is the first function, $v$ is second:</p>\n<p></p>\n<p style=\"text-align: center;\">$u=e^{\\var{a}x} $&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $v=\\ln{(\\var{b}x)}$</p>\n<p>&nbsp;</p>\n<p>Now, we need to use the approriate techniques to differentiate each of these, for these functions we need&nbsp;your<strong> <a href=\"resources/question-resources/Table_of_Derivatives.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Table of Derivatives</a></strong>:</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">This gives us:</p>\n<p style=\"text-align: center;\">$\\large \\frac{du}{dx}=\\var{a}e^{\\var{a}x} $&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $&nbsp;\\large \\frac{dv}{dx}= \\frac{1}{x}$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">We now use the formula:</p>\n<p style=\"text-align: center;\">$&nbsp;\\large &nbsp;\\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $</p>\n<p style=\"text-align: left;\">&nbsp;Make the appropriate substitutions into the formula:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\">$ \\large&nbsp; \\frac{dy}{dx}=&nbsp;e^{\\var{a}x} \\times \\frac{1}{x} + \\ln{(\\var{b}x)} \\times \\var{a}e^{\\var{a}x}&nbsp; $</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Finally, we need to use our basic algebra to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\">$ \\large&nbsp;\\frac{dy}{dx}= \\frac{ e^{\\var{a}x}}{x}&nbsp;+ \\var{a}e^{\\var{a}x} \\ln{(\\var{b}x)}&nbsp; &nbsp;$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">&nbsp;</p>", "rulesets": {}, "extensions": [], "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": "<h4>Differentiate&nbsp; $&nbsp; y=e^{\\var{a}x} \\ln{(\\var{b}x)}$</h4>\n<h4></h4>\n<p></p>\n<p style=\"text-align: center;\">$&nbsp; \\large \\frac{dy}{dx}=&nbsp; &nbsp;$[[0]]</p>\n<p style=\"text-align: center;\"></p>", "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": "(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", "type": "question", "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/"}]}