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{"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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Product Rule 03", "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></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>First identify the two functions&nbsp; $u$&nbsp; and&nbsp; $v$:</p>\n<p style=\"text-align: center;\">$u=$[[0]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $v=$[[1]]</p>\n<p>&nbsp;</p>\n<p>Now differentiate each one:</p>\n<p style=\"text-align: center;\">$&nbsp; \\large \\frac{du}{dx}=&nbsp; &nbsp;$[[2]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $&nbsp; \\large\\frac{dv}{dx}=&nbsp; &nbsp;$[[3]]</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Then using:</p>\n<p style=\"text-align: center;\">$&nbsp; \\large&nbsp;\\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx}&nbsp;$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Substitute each component into the formula in the correct place:</p>\n<p style=\"text-align: center;\">$&nbsp;&nbsp;\\large&nbsp;\\frac{dy}{dx}=$[[4]]$ &nbsp;\\large&nbsp;\\times$[[5]]$ \\large + $[[6]]$&nbsp;\\large&nbsp;\\times$[[7]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Finally tidy this up to give your final answer:</p>\n<p style=\"text-align: center;\">$&nbsp; \\large \\frac{dy}{dx}=&nbsp; &nbsp;$[[8]]</p>\n<p style=\"text-align: center;\"></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, 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