Hint:
\nNote 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: $\\displaystyle \\frac{d(xy)}{dx} = y+x\\frac{dy}{dx}$.
\nNow differentiate both sides of the relation with respect to $x$. Below is a worked solution to the problem.
\na) 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:
\\[\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x-{d}y) / ({b} + (2 * y)+{d}x)}\\]
\n\n", "name": "Implicit Differentiation", "preamble": {"js": "", "css": ""}, "variables": {"a": {"definition": "-random(2..9)", "name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "c": {"definition": "random(3..9 except -a+1)", "name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "b": {"definition": "c-1", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "d": {"definition": "random(-3..3 except[0,1])", "name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\tImplicit differentiation.
\n \t\tGiven $x^2+y^2+dxy +ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.
\n \t\tAlso 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"}, "parts": [{"sortAnswers": false, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "type": "jme", "failureRate": 1, "unitTests": [], "checkingType": "absdiff", "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}], "answer": "(({( - a)} + ( - (2 * x))-{d}y) / ({b} + (2 * y)+{d}x))", "vsetRange": [0, 1], "checkingAccuracy": 0.001, "notallowed": {"message": "
Input all numbers as integers or as fractions, not as decimals.
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\n$\\displaystyle \\frac{dy}{dx}= $ [[0]]
\nInput all numbers as integers not as decimals.
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\\[\\simplify{x^2+y^2+{d}x y+{a}x+{b}y}=\\var{c}\\]
where $y=y(x)$. Find $\\dfrac{dy}{dx}$.