// Numbas version: exam_results_page_options
{"name": "Harder implicit differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Harder implicit differentiation requiring both chain rule and product rule.</p>\n<p></p>\n<p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["c", "b", "a"], "rulesets": {}, "statement": "<p>Find $\\frac{dy}{dx}$ in terms of $x$ and $y$ using implicit differentiation in the following:</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>$\\var{a}xe^{\\var{b}y}=(x+y)^\\var{c}$</p>\n<p></p>\n<p>If we differentiate the left hand side with respect to x we obtain:</p>\n<p>$\\frac{d}{dx}(\\var{a}xe^{\\var{b}y}) = e^{\\var{b}y}\\frac{d}{dx}(\\var{a}x)+\\var{a}x \\frac{d}{dx}(e^{\\var{b}y})$ by the product rule</p>\n<p>$=\\var{a}e^{\\var{b}y}+\\var{a*b}xe^{\\var{b}y}\\frac{dy}{dx}$ by the&nbsp;chain rule</p>\n<p></p>\n<p>If we differentiate the&nbsp;right hand side with respect to x we obtain:</p>\n<p>$\\frac{d}{dx}(x+y)^\\var{c} = \\var{c}(x+y)^\\var{c-1}\\frac{d}{dx}(x+y)$ by the&nbsp;chain rule</p>\n<p>$=\\var{c}(x+y)^\\var{c-1}(1+\\frac{dy}{dx})$</p>\n<p></p>\n<p>So implicitly differentiating both sides of our original equation with respect to $x$ gives:</p>\n<p>$\\var{a}e^{\\var{b}y}+\\var{a*b}xe^{\\var{b}y}\\frac{dy}{dx}=\\var{c}(x+y)^\\var{c-1}(1+\\frac{dy}{dx})$</p>\n<p>Collecting $\\frac{dy}{dx}$ terms we obtain:</p>\n<p>$(\\var{a*b}xe^{\\var{b}y}-\\var{c}(x+y)^\\var{c-1})\\frac{dy}{dx}=\\var{c}(x+y)^\\var{c-1}-\\var{a}e^{\\var{b}y}$</p>\n<p>Hence</p>\n<p>$\\frac{dy}{dx}=\\frac{\\var{c}(x+y)^\\var{c-1}-\\var{a}e^{\\var{b}y}}{\\var{a*b}xe^{\\var{b}y}-\\var{c}(x+y)^\\var{c-1}}$</p>\n<p></p>", "extensions": [], "name": "Harder implicit differentiation", "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"c": {"description": "", "templateType": "anything", "name": "c", "definition": "random(3..7)", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "anything", "name": "a", "definition": "random(3..9)", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "name": "b", "definition": "random(-5..5 except 0 except 1 except -1)", "group": "Ungrouped variables"}}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "unitTests": [], "gaps": [{"expectedVariableNames": [], "answer": "({c}(x+y)^{c-1}-{a}e^({b}y))/(({a*b}x*e^({b}y)-{c}(x+y)^{c-1}))", "showCorrectAnswer": true, "showPreview": true, "marks": "4", "scripts": {}, "answerSimplification": "all,!collectNumbers", "type": "jme", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkVariableNames": false, "showFeedbackIcon": true, "checkingType": "absdiff", "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "checkingAccuracy": 0.001}], "sortAnswers": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p></p>\n<p>$\\var{a}xe^{\\var{b}y}=(x+y)^\\var{c}$</p>\n<p>$\\frac{dy}{dx}= $ [[0]]</p>", "customMarkingAlgorithm": ""}], "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}