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{"name": "Differentiation 5 - Trigonometric Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"p": {"templateType": "anything", "name": "p", "description": "", "definition": "repeat(random(3..6),2)", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "description": "<p>coefficients</p>", "definition": "shuffle(2..8)[0..5]", "group": "Ungrouped variables"}}, "statement": "<p><strong>Differentiate the following&nbsp;trigonometric functions using the chain rule.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>", "name": "Differentiation 5 - Trigonometric Functions", "parts": [{"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=\\sin(\\var{c[0]}x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": true, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "{c[0]}cos({c[0]}x)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=\\cos(\\var{c[1]}x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "-{c[1]}sin({c[1]}x)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=-\\sin(\\var{c[2]}x^2)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "-2{c[2]}x*cos({c[2]}x^2)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=-5\\cos(\\var{c[3]}x)+\\sin(\\var{c[4]}x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "5{c[3]}sin({c[3]}x)+{c[4]}cos({c[4]}x)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=\\tan^\\var{p[0]}(x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "sec", "value": ""}, {"name": "tan", "value": ""}, {"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "{p[0]}(tan^({p[0]}-1)(x))*sec^2(x)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$y=\\cos(x^\\var{p[1]}-1)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "-{p[1]}x^({p[1]}-1)*sin(x^{p[1]}-1)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}], "functions": {}, "rulesets": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "ungrouped_variables": ["c", "p"], "advice": "<p><strong></strong>If you don't know how to differentiate trigonometric functions, please see&nbsp;<em>'Differentiation 4 - Trigonometric Functions'</em>.</p>\n<p><strong></strong></p>\n<p><strong>These questions use the chain rule.</strong></p>\n<p>The earlier questions are easy to do by inspection, e.g using Part a:</p>\n<p style=\"text-align: center;\">$y=sin(\\var{c[0]}x)$.</p>\n<p>We differentiate the term(s) inside the function, here the term is $\\var{c[0]}x$.</p>\n<p>Then we derive $sin$ of any function, giving us $cos$.</p>\n<p>Putting our results together, we get</p>\n<p>$\\var{c[0]}cos(\\var{c[0]}x)$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong>We will now go through an entire worked example of the formal method of the chain rule using Part e.</strong></p>\n<p>The expression we will be differentiating here is</p>\n<p style=\"text-align: center;\">$y=tan^\\var{p[0]}(x)$.</p>\n<p style=\"text-align: left;\">As a reminder, the chain rule is defined as</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}$.</p>\n<p style=\"text-align: left;\">Now we let $u=tanx$, so then $y=u^\\var{p[0]}$</p>\n<p style=\"text-align: left;\">This becomes an easy differentiation using $\\frac{dy}{du}\\times\\frac{du}{dx}$:</p>\n<p style=\"text-align: left;\">Differentiate $y$ with respect to $u$, giving $\\simplify{{p[0]}u^{{p[0]}-1}}$.</p>\n<p style=\"text-align: left;\">Then differentiate $u$ with respect to $x$, giving $sec^2x$.</p>\n<p style=\"text-align: left;\">Multiply these results together, and substitue $tan$ back in for $u$.</p>\n<p style=\"text-align: left;\">Your final result is therefore</p>\n<p style=\"text-align: center;\">$\\simplify{{p[0]}(tan^{{p[0]}-1}(x))*sec^2(x)}$.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>More work on differentiation with&nbsp;trigonometric functions</p>"}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}