// Numbas version: exam_results_page_options
{"name": "Power Rule, Sum or Difference (Instructional)", "metadata": {"description": "<p>Fairly simple questions using differentiation \"power rule\" and \"sum or difference rules\" to differentiate single term functions and polynomials.</p>\n<p>Some co-efficients and indices can be negative and/or fractional.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "80", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Power Rule 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "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": {}, "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"}, {"name": "Power Rule 02", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "<p>Simple application of \"Power Rule\" to differentiate single term functions.</p>\n<p>Some co-efficients and powers are non-integer and 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[fractionNumbers]{n_2}} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{n_2} \\times \\var{a_2} x^{\\var[fractionNumbers]{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": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a_1": {"name": "a_1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "<p>Co-efficient for Q1</p>", "templateType": "anything", "can_override": false}, "n_1": {"name": "n_1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except 1 )", "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(fractional_powers )", "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}, "fractional_powers": {"name": "fractional_powers", "group": "Ungrouped variables", "definition": "[ 1/2, -1/2, 3/2, -3/2 ]", "description": "<p>List to allow random selection of fractional powers</p>", "templateType": "list of numbers", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a_1", "n_1", "a_2", "n_2", "a_3", "n_3", "fractional_powers"], "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[fractionNumbers]{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"}, {"name": "Power Rule 03", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "<p>Simple application of \"Power Rule\" to differentiate polynomials.</p>\n<p>Some co-efficients and powers are non-integer and 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 \"<em>Power Rule</em>\" 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;\">The \"<em>Sum or Difference Rules</em>\" also tell us that, <strong>as long as the terms are either added or subtracted</strong>, we can differentiate the function term by term.</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{a1} x^{\\var{n1}} + \\var{a2} x^{\\var{n2}} +\\var{a3} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\var{n1} \\times \\var{a1} x^{\\var{n1} - 1} + \\var{n2} \\times \\var{a2} x^{\\var{n2}-1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{a3} \\) )&nbsp; can be seen as &nbsp;\\( \\var{a3}x^0 \\) so will differentiate to \\( 0 \\times \\var{a3} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n1}*{a1}*x^({n1}-1) + {n2}*{a2}*x^({n2}-1) } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">b)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {b1} x^{{m1}} + {b2} x^{{m2}}+{b3}x^{{m3}}+{b4} }&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{m1} \\times \\var{b1} x^{\\var{m1} - 1} +\\var{m2} \\times \\var{b2} x^{\\var{m2} - 1}+\\var{m3} \\times \\var{b3} x^{\\var{m3} - 1}\\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{b4} \\) )&nbsp; can be seen as &nbsp;\\( \\var{b4}x^0 \\) so will differentiate to \\( 0 \\times \\var{b4} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\simplify{&nbsp; {m1}*{b1}*x^({m1}-1) + {m2}*{b2}*x^({m2}-1)+ {m3}*{b3}*x^({m3}-1)&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">c)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {c1} x^{{p1}} + {c2} x^{{p2}} } \\)</p>\n<p style=\"padding-left: 40px;\">rem \\( \\frac{dy}{dx } = \\var{p1} \\times \\var{c1} x^{\\var{p1} - 1}&nbsp; +\\var[fractionNumbers]{p2} \\times \\var{c2} x^{\\var[fractionNumbers]{p2} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{{p1}*{c1}*x^{{p1}-1} + {p2}*{c2}*x^{{p2}-1} } \\)</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>", 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except b1 except b2 except 0 )", "description": "", "templateType": "anything", "can_override": false}, "m3": {"name": "m3", "group": "Question B", "definition": "m2-1", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Question B", "definition": "random(-9..9 except b1 except b2 except b3 except 0 )", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Question C", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "p1": {"name": "p1", "group": "Question C", "definition": "random(-5..5 except 0 except 1)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Question C", "definition": "random(-9..9 except 0 except 1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], 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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>$ \\simplify{y= {b1} x^{{m1}} + {b2} x^{{m2}}+{b3}x^{{m3}}+{b4} }&nbsp; $</p>\n<p>\\( \\frac{dy}{dx } = \\) [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{m1}*{b1}*x^({m1}-1) + {m2}*{b2}*x^({m2}-1)+ {m3}*{b3}*x^({m3}-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$ \\simplify{y= {c1} x^{{p1}} + {c2} x^{{p2}} }&nbsp; $</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": "{p1}*{c1}*x^({p1}-1) + {p2}*{c2}*x^({p2}-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"}, {"name": "Power Rule 03", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "<p>Simple application of \"Power Rule\" to differentiate polynomials.</p>\n<p>Some co-efficients and powers are non-integer and 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 \"<em>Power Rule</em>\" 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;\">The \"<em>Sum or Difference Rules</em>\" also tell us that, <strong>as long as the terms are either added or subtracted</strong>, we can differentiate the function term by term.</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{a1} x^{\\var{n1}} + \\var{a2} x^{\\var{n2}} +\\var{a3} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\var{n1} \\times \\var{a1} x^{\\var{n1} - 1} + \\var{n2} \\times \\var{a2} x^{\\var{n2}-1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{a3} \\) )&nbsp; can be seen as &nbsp;\\( \\var{a3}x^0 \\) so will differentiate to \\( 0 \\times \\var{a3} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n1}*{a1}*x^({n1}-1) + {n2}*{a2}*x^({n2}-1) } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">b)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {b1} x^{{m1}} + {b2} x^{{m2}}+{b3}x^{{m3}}+{b4} }&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{m1} \\times \\var{b1} x^{\\var{m1} - 1} +\\var{m2} \\times \\var{b2} x^{\\var{m2} - 1}+\\var{m3} \\times \\var{b3} x^{\\var{m3} - 1}\\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{b4} \\) )&nbsp; can be seen as &nbsp;\\( \\var{b4}x^0 \\) so will differentiate to \\( 0 \\times \\var{b4} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\simplify{&nbsp; {m1}*{b1}*x^({m1}-1) + {m2}*{b2}*x^({m2}-1)+ {m3}*{b3}*x^({m3}-1)&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">c)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {c1} x^{{p1}} + {c2} x^{{p2}} } \\)</p>\n<p style=\"padding-left: 40px;\">rem \\( \\frac{dy}{dx } = \\var{p1} \\times \\var{c1} x^{\\var{p1} - 1}&nbsp; +\\var[fractionNumbers]{p2} \\times \\var{c2} x^{\\var[fractionNumbers]{p2} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{{p1}*{c1}*x^{{p1}-1} + {p2}*{c2}*x^{{p2}-1} } \\)</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p2": {"name": "p2", "group": "Question C", "definition": "random(fractional_powers )", "description": "", "templateType": "anything", "can_override": false}, "fractional_powers": {"name": "fractional_powers", "group": "Question C", "definition": "[ 1/2, -1/2, 3/2, -3/2, 1/3, -1/3, 2/3, -2/3, 1/4, -1/4, 3/4, -3/4 ]", "description": "<p>List to allow random selection of fractional powers</p>", "templateType": "list of numbers", "can_override": false}, "a1": {"name": "a1", "group": "Question A", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Question A", "definition": "random(3..6)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Question A", 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