// Numbas version: exam_results_page_options {"name": "Aoife's copy of Implicit Differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

You are given the following relation between $x$ and $y$
\$\\simplify{x^2+y^2+{d}x y+{a}x+{b}y}=\\var{c}\$
where $y=y(x)$. Find $\\dfrac{dy}{dx}$.

", "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers"]}, "extensions": [], "tags": [], "metadata": {"description": "\n \t\t

Implicit differentiation.

\n \t\t

Given $x^2+y^2+dxy +ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.

\n \t\t

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

\n \t\t

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "prompt": "

\n

$\\displaystyle \\frac{dy}{dx}=$ [[0]]

\n

Input all numbers as integers not as decimals.

", "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "gaps": [{"answerSimplification": "all,!collectNumbers", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "variableReplacements": [], "checkingType": "absdiff", "checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme", "checkingAccuracy": 0.001, "marks": 2, "showPreview": true, "expectedVariableNames": [], "showCorrectAnswer": true, "scripts": {}, "vsetRangePoints": 5, "answer": "(({( - a)} + ( - (2 * x))-{d}y) / ({b} + (2 * y)+{d}x))", "notallowed": {"partialCredit": 0, "showStrings": false, "strings": ["."], "message": "

Input all numbers as integers or as fractions, not as decimals.

"}, "failureRate": 1}]}], "ungrouped_variables": ["a", "c", "b", "d"], "advice": "

Hint:

\n

Note that we regard $y$ as a function of $x$. Hence we have (using the Chain Rule): $\\displaystyle \\frac{d(y^2)}{dx} = 2y\\frac{dy}{dx}$. And, using the Product Rule: $\\displaystyle \\frac{d(xy)}{dx} = y+x\\frac{dy}{dx}$.

\n

Now differentiate both sides of the relation with respect to $x$. Below is a worked solution to the problem, but only look at it if you are struggling.

\n

a) By differentiating both sides of the equation implicitly we get
\$2x + \\simplify[all,!collectNumbers]{2y*Diff(y,x,1) +{d}(y+x*Diff(y,x,1))+ {a} + {b} *Diff(y,x,1)} = 0\$
Collecting terms in $\\displaystyle\\frac{dy}{dx}$ and rearranging the equation we get
\$( \\simplify[all,!collectNumbers]{({b} + 2y+{d}x)} )\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{{ -a} -2x-{d}y}\$ and hence on further rearranging:

\n

\$\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x-{d}y) / ({b} + (2 * y)+{d}x)}\$

\n

\n

", "variables": {"a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "-random(2..9)"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "c-1"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..9 except -a+1)"}, "d": {"description": "", "name": "d", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except[0,1])"}}, "name": "Aoife's copy of Implicit Differentiation", "functions": {}, "variable_groups": [], "type": "question", "contributors": [{"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}, {"name": "Aoife O'Brien", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1693/"}, {"name": "Antonia Wilmot-Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2410/"}, {"name": "Leticija Dubickaite", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2461/"}]}]}], "contributors": [{"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}, {"name": "Aoife O'Brien", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1693/"}, {"name": "Antonia Wilmot-Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2410/"}, {"name": "Leticija Dubickaite", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2461/"}]}