// Numbas version: exam_results_page_options
{"name": "Power Rule 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Power Rule 01", "tags": [], "metadata": {"description": "<p>Simple application of \"Power Rule\" to differentiate single term functions.</p>\n<p>All co-efficients and powers are integer (though some may be negative.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>The Power Rule</h4>\n<p>You can find the derivative for powers of functions using the following rule:</p>\n<p style=\"text-align: center;\">If&nbsp; &nbsp;\\( y=ax^n \\)&nbsp; then&nbsp; &nbsp;\\( \\frac{dy}{dx} = n \\times a x^{n-1} \\)&nbsp;</p>\n<h4>The Sum or Difference Rules</h4>\n<p style=\"text-align: center;\">The derivative of&nbsp; &nbsp;\\( f(x) + g(x) \\)&nbsp; is&nbsp; \\(&nbsp; \\frac{df}{dx} + \\frac{dg}{dx} \\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; the derivative of&nbsp; &nbsp;\\( f(x) - g(x) \\)&nbsp; is&nbsp; \\(&nbsp; \\frac{df}{dx} - \\frac{dg}{dx} \\)</p>\n<p></p>", "advice": "<p>We are asked to differentiate a variety of functions, each consisting of a single term.</p>\n<p>We can do this using the \"Power Rule\" for differentiation:</p>\n<p style=\"text-align: center;\">If&nbsp; &nbsp;\\( y=ax^n \\)&nbsp; then&nbsp; &nbsp;\\( \\frac{dy}{dx} = n \\times a x^{n-1} \\)</p>\n<p style=\"text-align: left;\">In plain language, \"<em>multiply by the power, then reduce the power by one</em>\".</p>\n<p style=\"text-align: left;\">Then:</p>\n<p style=\"text-align: left;\">a)</p>\n<p style=\"padding-left: 40px;\">\\( y= \\var{a_1} x^{\\var{n_1}} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{n_1} \\times \\var{a_1} x^{\\var{n_1} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n_1}*{a_1}*x^{{n_1} - 1} } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">b)</p>\n<p style=\"padding-left: 40px;\">\\( y= \\var{a_2} x^{\\var{n_2}} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{n_2} \\times \\var{a_2} x^{\\var{n_2} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n_2}*{a_2}*x^{{n_2} - 1} } \\)</p>\n<p style=\"text-align: left;\">c)</p>\n<p style=\"padding-left: 40px;\">\\( y= \\var{a_3} x^{\\var{n_3}} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{n_3} \\times \\var{a_3} x^{\\var{n_3} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n_3}*{a_3}*x^{{n_3} - 1} } \\)</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a_1": {"name": "a_1", "group": "Ungrouped variables", "definition": "random(2 .. 9#1)", "description": "<p>Co-efficient for Q1</p>", "templateType": "randrange", "can_override": false}, "n_1": {"name": "n_1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "a_2": {"name": "a_2", "group": "Ungrouped variables", "definition": "random(3..15 except a_1 )", "description": "", "templateType": "anything", "can_override": false}, "n_2": {"name": "n_2", "group": "Ungrouped variables", "definition": "random(3..9 except n_1 )", "description": "", "templateType": "anything", "can_override": false}, "a_3": {"name": "a_3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except 1 )", "description": "", "templateType": "anything", "can_override": false}, "n_3": {"name": "n_3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except 1 )", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a_1", "n_1", "a_2", "n_2", "a_3", "n_3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Differentiate the following:</p>"}, {"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": "<p>\\( y= \\var{a_1} x^{\\var{n_1}} \\)</p>\n<p>\\( \\frac{dy}{dx } = \\)[[0]]</p>", "gaps": [{"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": "{n_1}*{a_1}*x^({n_1} - 1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"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": "<p>\\( y= \\var{a_2} x^{\\var{n_2}} \\)</p>\n<p>\\( \\frac{dy}{dx } = \\) [[0]]</p>", "gaps": [{"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": "{n_2}*{a_2}*x^({n_2} - 1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"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": "<p>\\( y= \\var{a_3} x^{\\var{n_3}} \\)</p>\n<p>\\( \\frac{dy}{dx } = \\)&nbsp; [[0]]</p>", "gaps": [{"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": "{n_3}*{a_3}*x^({n_3} - 1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": 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/"}]}