// Numbas version: exam_results_page_options
{"name": "Quotient Rule 05", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_W9at6dx.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_W9at6dx.pdf"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quotient Rule 05", "tags": [], "metadata": {"description": "<p>Instructional \"drill\" exercise to emphasize the method.</p><p>Thanks to Christian for his method for use of gaps in fractions.</p>", "licence": "All rights reserved"}, "statement": "<h4>We use the QUOTIENT RULE when the function that we need to differentiate is actually two functions divided:</h4>\n<h4></h4>\n<h4>If&nbsp; $\\large y=\\frac{u}{v}$&nbsp; then:</h4>\n<h4>\\[&nbsp; &nbsp;\\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} \\]</h4>\n<p></p>", "advice": "<p>We are asked to differentiate:</p>\n<p>\\[&nbsp;\\large&nbsp;y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}}&nbsp;\\]</p>\n<p></p>\n<p>Recognising that the function to differentiate is a quotient, we identify the two functions that are involved.</p>\n<p></p>\n<p>$u$ is the <strong><em>numerator</em></strong>, the function \"<em>on top</em>\", $v$ is the <em><strong>denominator</strong></em>, the function \"<em>on the bottom</em>\".</p>\n<p></p>\n<p style=\"text-align: center;\">$u=e^{\\var{aP}t}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $v=t+\\var{aC1}$</p>\n<p>&nbsp;</p>\n<p>Now, we need to use the approriate techniques to differentiate each of these, for $u$ we need the <b>Exponential Rule </b>and for $v$ we need the<b> Power Rule</b>:</p>\n<p></p>\n<p style=\"text-align: center;\">if&nbsp; $&nbsp; \\frac{d}{dx} [a^x] = a^x \\ln{(a)}$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">If you don't follow this, use your <a href=\"resources/question-resources/Table_of_Derivatives_W9at6dx.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Table of Derivatives</a>.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Applying this gives us:</p>\n<p style=\"text-align: center;\">$\\large \\frac{du}{dx}=\\simplify{{aP}e^({aP}t)}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\frac{dv}{dx}=1$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Make the appropriatesubstitutions into the formula:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\">$&nbsp;\\large&nbsp; \\frac{dy}{dx}= &nbsp;\\frac{(t+\\var{aC1}) \\times (\\simplify{{aP}e^({aP}t)}) - e^{\\var{aP}t} }{(t+\\var{aC1})^2}&nbsp; $</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Finally, we need to use our basic algebra terms to simplify this as much as possible. Multiply out brackets where it would simplify and collect like terms:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">$ \\large&nbsp;\\frac{dy}{dx}= \\frac{\\simplify{{aP}e^({aP}t)}(t+\\var{aC1}) - (e^{\\var{aP}t}) }{(t+\\var{aC1})^2}$&nbsp;</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\">&nbsp;</p>", "rulesets": {"std": ["all"]}, "extensions": [], "variables": {"aC1": {"name": "aC1", "group": "Part (a)", "definition": "random(2 .. 6#1)", "description": "<p>Part a) Constant term for&nbsp;denominator</p>\n<div id=\"gtx-trans\" style=\"position: absolute; left: 206px; top: 40px;\">\n<div class=\"gtx-trans-icon\"></div>\n</div>", "templateType": "randrange"}, "aP": {"name": "aP", "group": "Part (a)", "definition": "random(2..5 except aC1)", "description": "<p>Part a) x power for denominator</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (a)", "variables": ["aC1", "aP"]}], "functions": {}, "preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<h4>Differentiate&nbsp; $ \\large y=\\frac{e^{\\var{aP}t}}{t+\\var{aC1}}$</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;\">$&nbsp; \\large \\frac{du}{dt}=&nbsp; &nbsp;$[[2]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $&nbsp; \\large\\frac{dv}{dt}=&nbsp; &nbsp;$[[3]]</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Then using:</p>\n<p style=\"text-align: center;\">$&nbsp; \\Large \\frac{dy}{dx}= \\frac{v \\frac{du}{dx} - u \\frac{dv}{dx}}{v^{2}} $</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Substitute each component into the formula in the correct place:</p>\n<table style=\"margin-left: auto; margin-right: auto; float: left; width: 1147px;\">\n<tbody>\n<tr style=\"text-align: center;\">\n<td scope=\"row\" style=\"width: 168px;\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>\n<td scope=\"row\" style=\"width: 189px;\">$&nbsp;&nbsp;\\Large&nbsp;\\frac{dy}{dt}=$</td>\n<td scope=\"row\" style=\"width: 693px; text-align: left;\"><fraction><numerator></numerator></fraction><fraction><numerator>[[4]]$\\times$ [[5]]$\\large -$ [[6]]$\\times$ [[7]]</numerator><denominator>[[8]]</denominator></fraction></td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Finally tidy this up to give your final answer:</p>\n<p style=\"text-align: center;\">$&nbsp; \\Large \\frac{dy}{dx}=&nbsp; &nbsp;$ [[9]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "e^({aP}t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, 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"variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}e^({aP}t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "t+{aC1}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{aP}e^({aP}t)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "e^({aP}t)", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], 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"variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(t+{aC1})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({aP}e^({aP}t)*(t+{aC1} )-e^({aP}t))/(t+{aC1})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, 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