// Numbas version: exam_results_page_options
{"name": "Implicit differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-random(1..9)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Implicit differentiation", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "(({( - a)} + ( - (2 * x))) / ({b} + (2 * y)))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all,!collectNumbers", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "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      ", "showCorrectAnswer": true, "marks": 0}], "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>", "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"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "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"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}