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{"name": "CF Maths Portfolio 6 - Differentiation of Trigonometric Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c", "p"], "name": "CF Maths Portfolio 6 - Differentiation of Trigonometric Functions", "tags": [], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {}, "parts": [{"prompt": "<p>$y=-\\sin(\\var{c[2]}x^2)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-2{c[2]}x*cos({c[2]}x^2)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "5{c[3]}sin({c[3]}x)+{c[4]}cos({c[4]}x)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\tan^\\var{p[0]}(x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{p[0]}(tan(x))^({p[0]}-1)*(sec(x))^2", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\cos(x^\\var{p[1]}-1)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-{p[1]}x^({p[1]}-1)*sin(x^{p[1]}-1)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "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>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"p": {"definition": "repeat(random(3..6),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "c": {"definition": "shuffle(2..8)[0..5]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": "<p>coefficients</p>"}}, "metadata": {"description": "<p>More work on differentiation with&nbsp;trigonometric functions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}