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{"name": "Blathnaid's copy of Implicit differentiation", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "percentPass": 0, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Blathnaid's copy of Implicit differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b"], "tags": ["calculus", "Calculus", "checked2015", "derivative", "derivative ", "deriving an implicit relation", "differentiate", "differentiate implicitly", "differentiation", "first derivative using implicit differentiation", "implicit differentiation", "implicit relation", "mas1601", "MAS1601"], "type": "question", "advice": "<p>On differentiating both sides of the equation implicitly we get<br/>\\[2x + \\simplify[all,!collectNumbers]{2y*Diff(y,x,1) + {a} + {b} *Diff(y,x,1)} = 0\\]<br/>Collecting terms in $\\displaystyle\\frac{dy}{dx}$ and rearranging the equation we get <br/>\\[(\\var{b} + 2y) \\frac{dy}{dx} = \\simplify[all,!collectNumbers]{{ -a} -2x}\\] and hence on further rearranging:<br/>\\[\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x) / ({b} + (2 * y))}\\]</p>", "rulesets": {}, "parts": [{"prompt": "\n      <p>Using implicit differentiation find $\\displaystyle \\frac{dy}{dx}$&nbsp;in terms of $x$ and $y$.</p>\n        <p>Input your answer here:</p>\n        <p>$\\displaystyle \\frac{dy}{dx}= $ [[0]]</p>\n      ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "all,!collectNumbers", "scripts": {}, "answer": "(({( - a)} + ( - (2 * x))) / ({b} + (2 * y)))", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "preamble": {"css": "", "js": ""}, "statement": "<p>Given the following relation between $x$ and $y$<br/>\\[\\simplify[all,!collectNumbers]{x^2+y^2+{a}x+{b}y}=\\var{c}\\]<br/>answer the following question.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variables": {"a": {"definition": "-random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}}, "showQuestionGroupNames": false, "metadata": {"notes": "\n                            \t\t<p><strong>20/06/2012:</strong></p>\n                            \t\t<p>Added tags.</p>\n                            \t\t<p>Improved display using \\displaystyle where appropriate.</p>\n                            \t\t<p>Changed marks to 2.</p>\n                            \t\t<p>&nbsp;</p>\n                            \t\t<p><strong>3/07/2012:<br /></strong></p>\n                            \t\t<p>Added tags.</p>\n                            \t\t", "description": "\n                            \t\t<p>Implicit differentiation.</p>\n                            \t\t<p>Given $x^2+y^2+ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.</p>\n                            \t\t<p>&nbsp;</p>\n                            \t\t", "licence": "Creative Commons Attribution 4.0 International"}}]}], "showQuestionGroupNames": false, "metadata": {"description": "<p>Find $\\frac{\\mathrm{d}y}{\\mathrm{d}x}$ by differentiating an implicit equation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}], "extensions": [], "custom_part_types": [], "resources": []}