AP Calculus BC · Differentiation definitions and fundamental rules
The curve x^2+xy+y^2=7 in the xy-plane contains the point (2,1)
Problem
The curve \(x^2+xy+y^2=7\) in the \(xy\)-plane contains the point \((2,1)\). (a) Verify that \((2,1)\) lies on the curve. (b) Use implicit differentiation to find \(\dfrac{dy}{dx}\) in terms of \(x\) and \(y\). (c) Find the slope of the curve at \((2,1)\) and an equation of the tangent line to the curve at that point.
Hint
The middle term \(xy\) is a product; differentiate it with the product rule, not as if one factor were constant.
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