// Numbas version: exam_results_page_options {"name": "Differentiation - Trigonometric Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "parts": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": true, "answer": "{c[0]}cos({c[0]}x)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "
$y=\\sin(\\var{c[0]}x)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": false, "answer": "-{c[1]}sin({c[1]}x)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "$y=\\cos(\\var{c[1]}x)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": false, "answer": "-2{c[2]}x*cos({c[2]}x^2)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "$y=-\\sin(\\var{c[2]}x^2)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": false, "answer": "5{c[3]}sin({c[3]}x)+{c[4]}cos({c[4]}x)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "$y=-5\\cos(\\var{c[3]}x)+\\sin(\\var{c[4]}x)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "sec", "value": ""}, {"name": "tan", "value": ""}, {"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": false, "answer": "{p[0]}(tan^({p[0]}-1)(x))*sec^2(x)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "$y=\\tan^\\var{p[0]}(x)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": 0, "customName": "", "gaps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "failureRate": 1, "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "all", "checkVariableNames": false, "answer": "-{p[1]}x^({p[1]}-1)*sin(x^{p[1]}-1)", "adaptiveMarkingPenalty": 0, "type": "jme", "vsetRangePoints": 5, "unitTests": [], "checkingAccuracy": 0.001, "showFeedbackIcon": true, "checkingType": "absdiff", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "marks": "2", "customName": "", "vsetRange": [0, 1], "showPreview": true, "variableReplacements": [], "useCustomName": false}], "adaptiveMarkingPenalty": 0, "prompt": "$y=\\cos(x^\\var{p[1]}-1)$
\n$\\frac{dy}{dx}=$ [[0]]
", "showFeedbackIcon": true, "type": "gapfill", "useCustomName": false, "variableReplacements": [], "unitTests": []}], "statement": "Differentiate the following trigonometric functions using the chain rule.
\nDo not write out $dy/dx$; only input the differentiated right hand side of each equation.
", "advice": "If you don't know how to differentiate trigonometric functions, please see 'Differentiation 4 - Trigonometric Functions'.
\n\n
These questions use the chain rule.
\nThe earlier questions are easy to do by inspection, e.g using Part a:
\n$y=sin(\\var{c[0]}x)$.
\nWe differentiate the term(s) inside the function, here the term is $\\var{c[0]}x$.
\nThen we derive $sin$ of any function, giving us $cos$.
\nPutting our results together, we get
\n$\\var{c[0]}cos(\\var{c[0]}x)$.
\n\n\n\nWe will now go through an entire worked example of the formal method of the chain rule using Part e.
\nThe expression we will be differentiating here is
\n$y=tan^\\var{p[0]}(x)$.
\nAs a reminder, the chain rule is defined as
\n$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}$.
\nNow we let $u=tanx$, so then $y=u^\\var{p[0]}$
\nThis becomes an easy differentiation using $\\frac{dy}{du}\\times\\frac{du}{dx}$:
\nDifferentiate $y$ with respect to $u$, giving $\\simplify{{p[0]}u^{{p[0]}-1}}$.
\nThen differentiate $u$ with respect to $x$, giving $sec^2x$.
\nMultiply these results together, and substitue $tan$ back in for $u$.
\nYour final result is therefore
\n$\\simplify{{p[0]}(tan^{{p[0]}-1}(x))*sec^2(x)}$.
", "rulesets": {}, "ungrouped_variables": ["c", "p"], "extensions": [], "metadata": {"description": "More work on differentiation with trigonometric functions
", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Differentiation - Trigonometric Functions", "variable_groups": [], "preamble": {"css": "", "js": ""}, "variables": {"c": {"templateType": "anything", "name": "c", "description": "coefficients
", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..5]"}, "p": {"templateType": "anything", "name": "p", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(3..6),2)"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "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/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}]}], "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/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}