// Numbas version: finer_feedback_settings
{"name": "Differentiation: Implicit Differentiation 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Implicit Differentiation 4", "tags": [], "metadata": {"description": "<p>Calculating $\\frac{dy}{dx}$ from an implicit polynomial function.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If \\[ \\simplify{{a}x^{n}-{b}x*y-{c}y^{m}={d}}, \\]</p>\n<p>find $\\tfrac{dy}{dx}$.</p>", "advice": "<p>As we have an equation involving the variables $x$ and $y$, but cannot rearrange the equation into the form $y=f(x)$, we need to use implicit differentiation to find $\\tfrac{dy}{dx}$.&nbsp;</p>\n<p>Differentiating $\\simplify{{a}x^{n}-{b}x*y-{c}y^{m}={d}}$ with respect to $x$:</p>\n<p>\\[ \\simplify[all,!simplifyFractions,!cancelTerms]{{a*n}x^{n-1}-{b}y-{b}x} \\frac{dx}{dy}-\\simplify{{c*m}y^{m-1}}\\frac{dy}{dx} = 0.\\]</p>\n<p>Rearranging this equation so that it is in terms of $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\simplify{{-b}x} \\frac{dy}{dx}-\\simplify{{c*m}y^{m-1}} \\frac{dy}{dx} &amp;\\,= \\simplify[all,!noLeadingMinus]{{-a*n}x^{n-1}+{b}y}&nbsp; \\\\\\\\ -\\big(\\simplify{{b}x+{c*m}y^{m-1}}\\big)\\frac{dy}{dx} &amp;\\,=\\simplify[all,!noLeadingMinus]{{-a*n}x^{n-1}+{b}y} \\\\\\\\ \\big(\\simplify{{b}x+{c*m}y^{m-1}}\\big)\\frac{dy}{dx} &amp;\\,=\\simplify{{a*n}x^{n-1}-{b}y} \\\\\\\\ \\frac{dy}{dx} &amp;\\,= \\frac{\\simplify{{a*n}x^{n-1}-{b}y}}{\\simplify{{b}x+{c*m}y^{m-1}}} . \\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..6 except n)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..6 except b)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(a*n,b)=1 and gcd(c*m,b)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "m", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\dfrac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a*n}x^{n-1}-{b}y)/({b}x+{c*m}y^{m-1})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}