Linear approximation practice question 10

Differentiate implicitly to find $\dfrac{dy}{dx}$ at $(1, 2)$.

Differentiating both sides gives $3y^{2}y' + y + xy' = 0$; the $xy$ term needs the product rule. Solve for $y'$ and put in $(1, 2)$.

Then $y(1.1) \approx 2 + \dfrac{dy}{dx}\Big|_{(1,2)}\cdot(0.1)$.