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{"name": "Implicit Differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"a": {"description": "", "definition": "random(3..9)", "name": "a", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"description": "", "definition": "random(1..9)", "name": "b", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"description": "", "definition": "random(1..9)", "name": "c", "group": "Ungrouped variables", "templateType": "anything"}}, "functions": {}, "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "c", "b"], "name": "Implicit Differentiation", "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n    \t\t        \t\t                    \t\t<p>Implicit differentiation.</p>\n    \t\t        \t\t                    \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        \t\t                    \t\t<p>&nbsp;</p>\n    \t\t        \t\t                    \n    \t\t        \n    \t\t"}, "tags": ["calculus", "Calculus", "derivative", "deriving an implicit relation", "differentiate", "differentiate implicitly", "Differentiation", "differentiation", "first derivative using implicit differentiation", "implicit differentiation", "implicit relation"], "advice": "<p>\\[\\simplify[all,!collectNumbers]{x^2+y^2-{a}x*y+{b}y}=\\var{c}\\]</p>\n<p>On differentiating both sides of the equation implicitly we get<br/>\\[2x + \\simplify[all,!collectNumbers]{2y*Diff(y,x,1) - [{a}x*Diff(y,x,1)+{a}y] + {b} *Diff(y,x,1)} = 0\\]<br/>Collecting terms in $\\displaystyle\\dfrac{dy}{dx}$ and rearranging the equation we get <br/>\\[(\\var{b} -\\var{a}x+ 2y) \\dfrac{dy}{dx} = \\simplify[all,!collectNumbers]{{a}y -2x}\\] and hence on further rearranging:<br/>\\[\\frac{dy}{dx} = \\simplify{({a}y - 2 * x) / ({b} -{a}x +(2 * y))}\\]</p>", "parts": [{"showFeedbackIcon": true, "scripts": {}, "gaps": [{"checkingtype": "absdiff", "showFeedbackIcon": true, "scripts": {}, "marks": 2, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "vsetrange": [0, 1], "showCorrectAnswer": true, "answersimplification": "all,!collectNumbers", "showpreview": true, "checkvariablenames": false, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrangepoints": 5, "answer": "(({a}*y-2 * x) / ({b} + (2 * y)-{a}*x))"}], "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "<p>Input your answer here: &nbsp; $\\displaystyle \\frac{dy}{dx}= $ [[0]]</p>", "showCorrectAnswer": true}], "statement": "<p></p>\n<p>Given the&nbsp;<strong>implicit function:</strong></p>\n<p>\\[\\simplify[all,!collectNumbers]{x^2+y^2-{a}x*y+{b}y}=\\var{c}\\]</p>\n<p>find $\\displaystyle \\frac{dy}{dx}$&nbsp;in terms of $x$ and $y$, &nbsp;assuming that&nbsp;$y$ is a function of $x$.&nbsp;&nbsp;</p>\n<p>&nbsp;&nbsp;</p>", "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}